Via jupytext this file can be shown as a jupyter notebook.

)cd ..
)read input/jfricas-test-support.input )quiet

The current FriCAS default directory is /home/hemmecke/backup/git/qeta 
All user variables and function definitions have been cleared.
All )browse facility databases have been cleared.
Internally cached functions and constructors have been cleared.
 )clear completely is finished.
The current FriCAS default directory is /home/hemmecke/backup/git/qeta/tmp 

)set output algebra on
)set output formatted off

Problem formulation

Let $p(n)$ denote the number of partitions of the natural number $n$. Show that for every natural number $k$ the expression $p(11k+6)$ is divisible by 11.

The solution is described in the article (http://www.sciencedirect.com/science/article/pii/S0747717117300147):

@Article{Hemmecke_DancingSambaRamanujan_2018,
  author =       {Ralf Hemmecke},
  title =        {Dancing Samba with {R}amanujan Partition
                  Congruences},
  journal =      {Journal of Symbolic Computation,
  year =         2018,
  JournalUrl =   {http://www.sciencedirect.com/science/journal/07477171},
  URL =          {http://www.sciencedirect.com/science/article/pii/S0747717117300147},
  DOI =          {10.1016/j.jsc.2017.02.001},
  volume =       {84},
  pages =        {14--24},
  ISSN =         {0747-7171},
  keywords =     {Partition identities, Number theoretic algorithm,
                  Subalgebra basis},
  abstract =     {The article presents an algorithm to compute a
                  $C[t]$-module basis $G$ for a given subalgebra $A$
                  over a polynomial ring $R=C[x]$ with a Euclidean
                  domain $C$ as the domain of coefficients and $t$ a
                  given element of $A$. The reduction modulo $G$
                  allows a subalgebra membership test. The algorithm
                  also works for more general rings $R$, in particular
                  for a ring $R\subset C((q))$ with the property that
                  $f\in R$ is zero if and only if the order of $f$ is
                  positive. As an application, we algorithmically
                  derive an explicit identity (in terms of quotients
                  of Dedekind eta-functions and Klein's
                  $j$-invariant) that shows that $p(11n+6)$ is
                  divisible by 11 for every natural number $n$ where
                  $p(n)$ denotes the number of partitions of $n$.}
 }

The article is also available as RISC report 16-06.

Explanation of the session

Let X be an algebra that can be used with the algorithm samba, but which has a polynomial attached to record the operations that have been done with the original element. This somehow simulates a similar idea as attaching a unit matrix to a matrix and then computing the row echelon form of the matrix. The unit matrix turns into the transformation matrix.

Since we want to show divisibility by 11 of $p(11k+6)$ for every $k$ where $p$ is the partition function, the main computation is done with a coefficient ring $C=\mathbb{Z}_{(11)}$ being integers localized at 11, i.e. rational numbers that have no denominator divisible by 11.

-------------------------------------------------------------------
--setup
-------------------------------------------------------------------

C ==> IntegerLocalizedAtPrime 11
L ==> A1 C
X ==> X1 C
ORBIT ==> modularOrbit $ QMOD0

PolC ==> Pol C; LC ==> A1 C
PolZ ==> Pol ZZ; LZZ ==> A1 ZZ
PolQ ==> Pol QQ; LQQ ==> A1 QQ

smaller11(x: PolC, y: PolC):Boolean == _
  exponent leadingCoefficient x < exponent leadingCoefficient y
smallerT(x: PolC, y: PolC):Boolean == degree(x, 'T) < degree(y, 'T)

Function declaration smaller11 : (Polynomial(IntegerLocalizedAtPrime(11)), 
Polynomial(IntegerLocalizedAtPrime(11))) -> Boolean has been added to 
workspace.
Function declaration smallerT : (Polynomial(IntegerLocalizedAtPrime(11)), 
Polynomial(IntegerLocalizedAtPrime(11))) -> Boolean has been added to 
workspace.

)set mess type on
)set mess time on

-------------------------------------------------------------------
--endsetup
-------------------------------------------------------------------

Relations in terms of the $M_i$'s

-------------------------------------------------------------------
--test:Hemmecke
-------------------------------------------------------------------

The generators of the eta-quotients of level 22 are given by these exponents for the eta-functions (with arguments being the divisors of 22). The variable rgens corresponds to the generators of $R^\infty(22)$.

nn := 22;
idxs := etaFunctionIndices nn
mspecs := mSPECSInfM0(nn, idxs);
rgensExpected: List List ZZ := [_
    [- 4, 8, 4, - 8], [- 1, 1, 11, - 11], [7, - 3, 3, - 7],_
    [- 2, 6, 6, - 10], [0, 4, 8, - 12], [2, 2, 10, - 14], [4, 0, 12, - 16]];
rgens := [pureExponents x for x in mspecs]
assertEquals(rgens, rgensExpected)

_{PositiveInteger}

_{Time: 0 sec}

\[ \left[\left[1\right], \left[2\right], \left[11\right], \left[22\right]\right] \]

_{List(List(Integer))}

_{Time: 0.01 (OT) = 0.01 sec}

_{List(QEtaSpecification)}

_{Time: 0.01 (EV) + 0.02 (OT) = 0.03 sec}

_{List(List(Integer))}

_{Time: 0 sec}

\[ \left[\left[-4, 8, 4, -8\right], \left[-1, 1, 11, -11\right], \left[7, -3, 3, -7\right], \left[-2, 6, 6, -10\right], \left[0, 4, 8, -12\right], \left[2, 2, 10, -14\right], \left[4, 0, 12, -16\right]\right] \]

_{List(List(Integer))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

eqgensZ := [specEQI(ZZ)(b) for b in mspecs]
assertEquals(removeDuplicates [one? denom eulerExponent x for x in eqgensZ], [true])

\[ \left[{q}^{-5}\, \left(1+4\, q+6\, {q}^{2}+8\, {q}^{3}+13\, {q}^{4}+12\, {q}^{5}+14\, {q}^{6}+24\, {q}^{7}+18\, {q}^{8}+20\, {q}^{9}+32\, {q}^{10}+O\left({q}^{11}\right)\right), {q}^{-5}\, \left(1+q+{q}^{2}+2\, {q}^{3}+2\, {q}^{4}+3\, {q}^{5}+4\, {q}^{6}+5\, {q}^{7}+6\, {q}^{8}+8\, {q}^{9}+10\, {q}^{10}+O\left({q}^{11}\right)\right), {q}^{-5}\, \left(1-7\, q+17\, {q}^{2}-14\, {q}^{3}+2\, {q}^{4}-21\, {q}^{5}+36\, {q}^{6}+13\, {q}^{7}-26\, {q}^{8}-24\, {q}^{9}+10\, {q}^{10}+O\left({q}^{11}\right)\right), {q}^{-6}\, \left(1+2\, q-{q}^{2}-2\, {q}^{3}-{q}^{4}-6\, {q}^{5}+4\, {q}^{7}-5\, {q}^{8}+4\, {q}^{9}+6\, {q}^{10}+O\left({q}^{11}\right)\right), {q}^{-7}\, \left(1-4\, {q}^{2}+2\, {q}^{4}+8\, {q}^{6}-5\, {q}^{8}-4\, {q}^{10}+O\left({q}^{11}\right)\right), {q}^{-8}\, \left(1-2\, q-3\, {q}^{2}+6\, {q}^{3}+2\, {q}^{4}-{q}^{6}-10\, {q}^{7}-2\, {q}^{9}+10\, {q}^{10}+O\left({q}^{11}\right)\right), {q}^{-9}\, \left(1-4\, q+2\, {q}^{2}+8\, {q}^{3}-5\, {q}^{4}-4\, {q}^{5}-10\, {q}^{6}+8\, {q}^{7}+9\, {q}^{8}+14\, {q}^{10}+O\left({q}^{11}\right)\right)\right] \]

_{List(EtaQuotientQSeriesInfinity(Integer))}

_{Time: 0.02 (OT) = 0.03 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

egensZ := [expansion x for x in eqgensZ];
rfegensZ := [_
  (12*q^5+13*q^4+8*q^3+6*q^2+4*q+1)/q^5,_
  (3*q^5+2*q^4+2*q^3+q^2+q+1)/q^5,_
  (-21*q^5+2*q^4-14*q^3+17*q^2-7*q+1)/q^5,_
  (-6*q^5-q^4-2*q^3-q^2+2*q+1)/q^6,_
  (8*q^6+2*q^4-4*q^2+1)/q^7,_
  (-10*q^7-q^6+2*q^4+6*q^3-3*q^2-2*q+1)/q^8,_
  (9*q^8+8*q^7-10*q^6-4*q^5-5*q^4+8*q^3+2*q^2-4*q+1)/q^9];
assertEquals([rationalFunction(x, 0) for x in egensZ], rfegensZ)

_{List(QEtaLaurentSeries(Integer))}

_{Time: 0 sec}

_{List(Fraction(Polynomial(Integer)))}

_{Time: 0.01 (OT) = 0.02 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

egens := [specM0A1(C)(x) for x in mspecs];
assertEquals([qetaGrade x for x in egens], [5, 5, 5, 6, 7, 8, 9])

_{List(ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

msyms := indexedSymbols("M", #egens)
assertEquals(msyms, [M1, M2, M3, M4, M5, M6, M7])

\[ \left[M1, M2, M3, M4, M5, M6, M7\right] \]

_List(Symbol)

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

xgens := [toX1(C, x, s) for x in egens for s in msyms]; -- List X

_{List(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0 sec}

We take a minimal (in terms the order in $q$) element from the generators and make it special. Then we compute an algebra basis for $C[M_1,\ldots,M_7]$.

19-Feb-2020: Interestingly, it does not work for the element corresponding to $[- 1, 1, 11, - 11]$. We choose the vector corresponding to $[- 4, 8, 4, - 8]$.

xab := samba(xgens)$QSAMBA(C,X1,QTOPRED)

-- numOfGaps:=[216, -1]
-- numOfGaps:=[216, -1]
-- numOfGaps:=[162, -1]
-- numOfGaps:=[162, -1]
-- numOfGaps:=[109, -1]
-- numOfGaps:=[109, -1]
-- numOfGaps:=[56, -1]
-- numOfGaps:=[56, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]

\[ \left[mul=\left[{q}^{-5}+4\, {q}^{-4}+6\, {q}^{-3}+8\, {q}^{-2}+13\, {q}^{-1}+12+14\, q+24\, {q}^{2}+18\, {q}^{3}+20\, {q}^{4}+32\, {q}^{5}+O\left({q}^{6}\right), M1\right], be=\texttt{table}\left(4=\left[\left[{q}^{-4}+\frac{5}{3}\, {q}^{-3}+2\, {q}^{-2}+11\, \frac{1}{3}\, {q}^{-1}+3+\frac{10}{3}\, q+\frac{19}{3}\, {q}^{2}+4\, {q}^{3}+4\, {q}^{4}+11\, \frac{2}{3}\, {q}^{5}+\frac{19}{3}\, {q}^{6}+O\left({q}^{7}\right), -\frac{1}{3}\, M2+\frac{1}{3}\, M1\right]\right], 3=\left[\left[11\, {q}^{-3}+11\, {q}^{-1}+11\, 2\, q+11\, 2\, {q}^{2}+11\, 2\, {q}^{5}+11\, 2\, {q}^{6}+11\, {q}^{7}+O\left({q}^{8}\right), \frac{3}{8}\, M3+11\, \left(-\frac{1}{8}\right)\, M2+M1\right], \left[{q}^{-8}-2\, {q}^{-7}-3\, {q}^{-6}+6\, {q}^{-5}+2\, {q}^{-4}-{q}^{-2}-10\, {q}^{-1}-2\, q+10\, {q}^{2}+O\left({q}^{3}\right), M6\right]\right], 2=\left[\left[{q}^{-7}-4\, {q}^{-5}+2\, {q}^{-3}+8\, {q}^{-1}-5\, q-4\, {q}^{3}+O\left({q}^{4}\right), M5\right]\right], 1=\left[\left[{q}^{-6}+2\, {q}^{-5}-{q}^{-4}-2\, {q}^{-3}-{q}^{-2}-6\, {q}^{-1}+4\, q-5\, {q}^{2}+4\, {q}^{3}+6\, {q}^{4}+O\left({q}^{5}\right), M4\right]\right]\right)\right] \]

_{QEtaAlgebraBasis(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0.01 (EV) + 0.01 (OT) = 0.03 sec}

Due to another critical element selection strategy, the resulting polynomial is a bit different.

bas := sort(smallerGrade?, basis xab);
xabExpectedArticle := [1/128*M7-5/128*M6+21/128*M5_
  +(-1/1024*M1+11*213/1024)*M4-85/128*M3+1/1024*M1^2+505/1024*M1,_
  1/8*M4-1/8*M1, M3, M5, M6] -- LPol C
xabExpected := [3/8*M3+11*(-1/8)*M2+M1, -1/3*M2+1/3*M1, M4, M5, M6]
assertEquals([x1Pol(C,x) for x in bas], xabExpected)
assertEquals(# bas, 5)

Compiling function smallerGrade? with type (QEtaExtendedAlgebra(
IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(
IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),
Polynomial(IntegerLocalizedAtPrime(11)))), QEtaExtendedAlgebra(
IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(
IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),
Polynomial(IntegerLocalizedAtPrime(11))))) -> Boolean 

_{List(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0 sec}

\[ \left[\frac{1}{128}\, M7-\frac{5}{128}\, M6+\frac{21}{128}\, M5+\left(-\frac{1}{1024}\, M1+\frac{2343}{1024}\right)\, M4-\frac{85}{128}\, M3+\frac{1}{1024}\, {M1}^{2}+\frac{505}{1024}\, M1, \frac{1}{8}\, M4-\frac{1}{8}\, M1, M3, M5, M6\right] \]

_{List(Polynomial(Fraction(Integer)))}

_{Time: 0.01 sec}

\[ \left[\frac{3}{8}\, M3-\frac{11}{8}\, M2+M1, -\frac{1}{3}\, M2+\frac{1}{3}\, M1, M4, M5, M6\right] \]

_{List(Polynomial(Fraction(Integer)))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

Relation for divisibility of $p(11k+6)$ by 11

Let's first define $F$ as on top of page 30 of \cite{Radu_RamanujanKolberg_2015}, i.e. $$ F(\tau) = q^{\frac{13}{24}} \frac{\eta(\tau)^{10}\eta(2\tau)^2\eta(11\tau)^{11}} {\eta(22\tau)^{22}} \sum_{k=0}^{\infty}p(11k+6)q^k $$

First we define the partion series.

ps := partitionSeries(1)$QFunctions(ZZ, LZZ) --: LZZ

\[ 1+q+2\, {q}^{2}+3\, {q}^{3}+5\, {q}^{4}+7\, {q}^{5}+11\, {q}^{6}+15\, {q}^{7}+22\, {q}^{8}+30\, {q}^{9}+42\, {q}^{10}+O\left({q}^{11}\right) \]

_{ModularFunctionQSeriesInfinity(Integer)}

_{Time: 0 sec}

Then we select the sub-series given by the (11n+6)-th term.

r11 := choose(11, 6, ps) --: LZZ
r11Expected := _
  1188908248*q^10+342325709*q^9+92669720*q^8+23338469*q^7+5392783*q^6_
  +1121505*q^5+204226*q^4+31185*q^3+3718*q^2+297*q+11 --: Fraction PolZ
assertEquals(rationalFunction(r11, 10), r11Expected)

\[ 11+297\, q+3718\, {q}^{2}+31185\, {q}^{3}+204226\, {q}^{4}+1121505\, {q}^{5}+5392783\, {q}^{6}+23338469\, {q}^{7}+92669720\, {q}^{8}+342325709\, {q}^{9}+1188908248\, {q}^{10}+O\left({q}^{11}\right) \]

_{ModularFunctionQSeriesInfinity(Integer)}

_{Time: 0 sec}

\[ 1188908248\, {q}^{10}+342325709\, {q}^{9}+92669720\, {q}^{8}+23338469\, {q}^{7}+5392783\, {q}^{6}+1121505\, {q}^{5}+204226\, {q}^{4}+31185\, {q}^{3}+3718\, {q}^{2}+297\, q+11 \]

_{Polynomial(Integer)}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

rspec := eqSPEC(1,[-1]); nn := 22; m := 11; t := 6;
sspec := cofactInfM0(nn, idxs, rspec, m, t)
assertEquals(sspec, eqSPEC(nn,[10,2,11,-22]))

_{PositiveInteger}

_{Time: 0 sec}

-- == z:=[zinhom=[], zhom=[], zfree=[]]
-- >= z:=[zinhom=[[4, -1, 1], [5, -1, 2], [4, -1, 2]], zhom=[[3, -1, 0], [4, -1, 1], [1, -1, 1], [2, -1, 1], [3, -1, 1], [0, -1, 1], [2, 0, 1]], zfree=[]]

\[ \left[\left[1, 10\right], \left[2, 2\right], \left[11, 11\right], \left[22, -22\right]\right] \]

_{QEtaSpecification}

_{Time: 0.07 (EV) + 0.02 (OT) = 0.09 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

fZ := specM0A1(ZZ)(sspec,rspec,m,t)
fZExpected := _
  1272909*q^14+944328*q^13+689282*q^12+485254*q^11+327415*q^10+214500*q^9+_
  135608*q^8+79211*q^7+44396*q^6+22286*q^5+10164*q^4+3696*q^3+1111*q^2+187*q+11
  --: PolZ
assertEquals(numer rationalFunction(fZ, 0), fZExpected)
assertEquals(qetaGrade fZ, 14)

\[ 11\, {q}^{-14}+187\, {q}^{-13}+1111\, {q}^{-12}+3696\, {q}^{-11}+10164\, {q}^{-10}+22286\, {q}^{-9}+44396\, {q}^{-8}+79211\, {q}^{-7}+135608\, {q}^{-6}+214500\, {q}^{-5}+327415\, {q}^{-4}+O\left({q}^{-3}\right) \]

_{ModularFunctionQSeriesInfinity(Integer)}

_{Time: 0 sec}

\[ 1272909\, {q}^{14}+944328\, {q}^{13}+689282\, {q}^{12}+485254\, {q}^{11}+327415\, {q}^{10}+214500\, {q}^{9}+135608\, {q}^{8}+79211\, {q}^{7}+44396\, {q}^{6}+22286\, {q}^{5}+10164\, {q}^{4}+3696\, {q}^{3}+1111\, {q}^{2}+187\, q+11 \]

_{Polynomial(Integer)}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

xf := toX1(C, specM0A1(C)(sspec,rspec,m,t), F) --: X

\[ \left[11\, {q}^{-14}+11\, 17\, {q}^{-13}+11\, 101\, {q}^{-12}+11\, 336\, {q}^{-11}+{11}^{2}\, 84\, {q}^{-10}+11\, 2026\, {q}^{-9}+11\, 4036\, {q}^{-8}+11\, 7201\, {q}^{-7}+11\, 12328\, {q}^{-6}+11\, 19500\, {q}^{-5}+11\, 29765\, {q}^{-4}+O\left({q}^{-3}\right), F\right] \]

_{QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11))))}

_{Time: 0 sec}

Now we can reduce this element xf by the respective algebra basis, taking into account that xt=xgens.1 is our special $t$ that we used in the construction of the algebra basis of the eta-quotients of level 22.

As can be seen from the sorted monomials of the representaion, $F$ can be expressed as $F=11p(M_1,\ldots, M_7)$ where $p$ is a polynomial having coefficient denominators not divisible by 11.

xr := reduce(xf, xab)$QTOPRED(C,X1)
assertTrue(zero? xr)

\[ \left[O\left({q}^{22}\right), \left(11\, M1+{11}^{3}\, 7\right)\, M5+\left(11\, 3\, M1+{11}^{3}\, 5\right)\, M4+\left(11\, \left(-\frac{1}{4}\right)\, {M1}^{2}+11\, \left(-\frac{245}{8}\right)\, M1+{11}^{3}\, \left(-\frac{1}{8}\right)\right)\, M3+\left(11\, \frac{5}{4}\, {M1}^{2}+{11}^{2}\, \left(-\frac{49}{8}\right)\, M1+{11}^{5}\, \frac{1}{8}\right)\, M2+11\, \left(-1\right)\, {M1}^{3}+11\, 3\, {M1}^{2}+{11}^{2}\, \left(-13\right)\, M1+F+{11}^{4}\, \left(-1\right)\right] \]

_{QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11))))}

_{Time: 0.01 (OT) = 0.02 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

Due to a different reduction strategy in a newer implementation of Samba, we don't get exactly the same result as in \cite{Hemmecke_DancingSambaRamanujan_2018}.

mxr := [(inv(unitPart leadingCoefficient x)::C)*x _
        for x in monomials(x1Pol(C,xr))]
mxrExpectedArticle := _
  [F, 11*M1^2, 11*M1^3, 11*M1*M3, 11*M1^2*M4, 11*M1*M5, 11^2*M1, 11^2*M3,_
  11^2*M1*M4, 11^2*M5, 11^2*M1*M6, 11^2*M6, 11^2*M7, 11^3, 11^3*M4]$LPol(C)
mxrExpected := _
  [F, 11*M1^2, 11*M1^3, 11*M1^2*M2, 11*M1*M3, 11*M1^2*M3, 11*M1*M4, _
  11*M1*M5, 11^2*M1, 11^2*M1*M2, 11^3*M3, 11^3*M4, 11^3*M5, 11^4, 11^5*M2]$LPol(C)
assertEquals(sort(smaller11, mxr), mxrExpected)

\[ \left[11\, M1\, M5, {11}^{3}\, M5, 11\, M1\, M4, {11}^{3}\, M4, 11\, {M1}^{2}\, M3, 11\, M1\, M3, {11}^{3}\, M3, 11\, {M1}^{2}\, M2, {11}^{2}\, M1\, M2, {11}^{5}\, M2, 11\, {M1}^{3}, 11\, {M1}^{2}, {11}^{2}\, M1, F, {11}^{4}\right] \]

_{List(Polynomial(IntegerLocalizedAtPrime(11)))}

_{Time: 0 sec}

\[ \left[F, 11\, {M1}^{2}, 11\, {M1}^{3}, 11\, M1\, M3, 11\, {M1}^{2}\, M4, 11\, M1\, M5, {11}^{2}\, M1, {11}^{2}\, M3, {11}^{2}\, M1\, M4, {11}^{2}\, M5, {11}^{2}\, M1\, M6, {11}^{2}\, M6, {11}^{2}\, M7, {11}^{3}, {11}^{3}\, M4\right] \]

_{List(Polynomial(IntegerLocalizedAtPrime(11)))}

_{Time: 0.01 sec}

\[ \left[F, 11\, {M1}^{2}, 11\, {M1}^{3}, 11\, {M1}^{2}\, M2, 11\, M1\, M3, 11\, {M1}^{2}\, M3, 11\, M1\, M4, 11\, M1\, M5, {11}^{2}\, M1, {11}^{2}\, M1\, M2, {11}^{3}\, M3, {11}^{3}\, M4, {11}^{3}\, M5, {11}^{4}, {11}^{5}\, M2\right] \]

_{List(Polynomial(IntegerLocalizedAtPrime(11)))}

_{Time: 0 sec}

Compiling function smaller11 with type (Polynomial(IntegerLocalizedAtPrime(11
)), Polynomial(IntegerLocalizedAtPrime(11))) -> Boolean 

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

Let's normalize the leading coefficient of $F$ to 1.

xrq := x1Pol(C,xr);
cxfq := coefficient(xrq, 'F, 1)
lcxfq := leadingCoefficient cxfq
lcfxqArticle := -1/1360
assertEquals(lcxfq, 1)
ilcxfq := (inv lcxfq)::C
xrq := ilcxfq * xrq
xrqExpectedArticle := _
  (11^2*(-3068)*M7+(11^2*(-3)*M1+11^2*(-4236))*M6+(11*(-285)*M1_
  +11^2*(-5972))*M5+(11*(-1/8)*M1^2+11^2*(-4497/8)*M1+11^3*(-789/2))*M4_
  +(11*(-1867)*M1+11^2*(-2476))*M3+11*1/8*M1^3+11*1011/8*M1^2_
  +11^2*1647/2*M1+F+11^3*1360)$PolC
xrqExpected := (11*(_
  (M1+11^2*7)*M5 _
  +(3*M1+11^2*5)*M4 _
  +((-1/4)*M1^2 + (-245/8)*M1+11^2*(-1/8))*M3 _
  +(5/4*M1^2+11*(-49/8)*M1+11^4*1/8)*M2 _
  +(-1)*M1^3+3*M1^2+11*(-13)*M1+11^3*(-1)) _
  +F)$PolC
assertEquals(xrq, xrqExpected)

_{Polynomial(IntegerLocalizedAtPrime(11))}

_{Time: 0 sec}

\[ 1 \]

_{Polynomial(IntegerLocalizedAtPrime(11))}

_{Time: 0 sec}

\[ 1 \]

_{IntegerLocalizedAtPrime(11)}

_{Time: 0 sec}

\[ -\frac{1}{1360} \]

_{Fraction(Integer)}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

\[ 1 \]

_{IntegerLocalizedAtPrime(11)}

_{Time: 0 sec}

\[ \left(11\, M1+{11}^{3}\, 7\right)\, M5+\left(11\, 3\, M1+{11}^{3}\, 5\right)\, M4+\left(11\, \left(-\frac{1}{4}\right)\, {M1}^{2}+11\, \left(-\frac{245}{8}\right)\, M1+{11}^{3}\, \left(-\frac{1}{8}\right)\right)\, M3+\left(11\, \frac{5}{4}\, {M1}^{2}+{11}^{2}\, \left(-\frac{49}{8}\right)\, M1+{11}^{5}\, \frac{1}{8}\right)\, M2+11\, \left(-1\right)\, {M1}^{3}+11\, 3\, {M1}^{2}+{11}^{2}\, \left(-13\right)\, M1+F+{11}^{4}\, \left(-1\right) \]

_{Polynomial(IntegerLocalizedAtPrime(11))}

_{Time: 0 sec}

\[ {11}^{2}\, \left(-3068\right)\, M7+\left({11}^{2}\, \left(-3\right)\, M1+{11}^{2}\, \left(-4236\right)\right)\, M6+\left(11\, \left(-285\right)\, M1+{11}^{2}\, \left(-5972\right)\right)\, M5+\left(11\, \left(-\frac{1}{8}\right)\, {M1}^{2}+{11}^{2}\, \left(-\frac{4497}{8}\right)\, M1+{11}^{3}\, \left(-\frac{789}{2}\right)\right)\, M4+\left(11\, \left(-1867\right)\, M1+{11}^{2}\, \left(-2476\right)\right)\, M3+11\, \frac{1}{8}\, {M1}^{3}+11\, \frac{1011}{8}\, {M1}^{2}+{11}^{2}\, \frac{1647}{2}\, M1+F+{11}^{3}\, 1360 \]

_{Polynomial(IntegerLocalizedAtPrime(11))}

_{Time: 0.02 (IN) = 0.03 sec}

\[ \left(11\, M1+{11}^{3}\, 7\right)\, M5+\left(11\, 3\, M1+{11}^{3}\, 5\right)\, M4+\left(11\, \left(-\frac{1}{4}\right)\, {M1}^{2}+11\, \left(-\frac{245}{8}\right)\, M1+{11}^{3}\, \left(-\frac{1}{8}\right)\right)\, M3+\left(11\, \frac{5}{4}\, {M1}^{2}+{11}^{2}\, \left(-\frac{49}{8}\right)\, M1+{11}^{5}\, \frac{1}{8}\right)\, M2+11\, \left(-1\right)\, {M1}^{3}+11\, 3\, {M1}^{2}+{11}^{2}\, \left(-13\right)\, M1+F+{11}^{4}\, \left(-1\right) \]

_{Polynomial(IntegerLocalizedAtPrime(11))}

_{Time: 0.01 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

Let's plug in the original eta-quotients into xrq.

sers := cons(first xf, egens) --: List L
-- Note that after some changes to the sorting order in the
-- implementation, the old Mi correspond to other new Mi.
sersArticle:=cons(first xf, [egens.i for i in [3,1,4,2,5,6,7]])
ezArticle := eval(xrqExpectedArticle, c+->c::L, cons('F,msyms), sersArticle)_
      $ PolynomialEvaluation(C, L)
assertTrue(zero? ezArticle)
ez := eval(xrq, c+->c::L, cons('F,msyms), sers)_
      $ PolynomialEvaluation(C, L)
assertTrue(zero? ez)

\[ \left[11\, {q}^{-14}+11\, 17\, {q}^{-13}+11\, 101\, {q}^{-12}+11\, 336\, {q}^{-11}+{11}^{2}\, 84\, {q}^{-10}+11\, 2026\, {q}^{-9}+11\, 4036\, {q}^{-8}+11\, 7201\, {q}^{-7}+11\, 12328\, {q}^{-6}+11\, 19500\, {q}^{-5}+11\, 29765\, {q}^{-4}+O\left({q}^{-3}\right), {q}^{-5}+4\, {q}^{-4}+6\, {q}^{-3}+8\, {q}^{-2}+13\, {q}^{-1}+12+14\, q+24\, {q}^{2}+18\, {q}^{3}+20\, {q}^{4}+32\, {q}^{5}+O\left({q}^{6}\right), {q}^{-5}+{q}^{-4}+{q}^{-3}+2\, {q}^{-2}+2\, {q}^{-1}+3+4\, q+5\, {q}^{2}+6\, {q}^{3}+8\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), {q}^{-5}-7\, {q}^{-4}+17\, {q}^{-3}-14\, {q}^{-2}+2\, {q}^{-1}-21+36\, q+13\, {q}^{2}-26\, {q}^{3}-24\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), {q}^{-6}+2\, {q}^{-5}-{q}^{-4}-2\, {q}^{-3}-{q}^{-2}-6\, {q}^{-1}+4\, q-5\, {q}^{2}+4\, {q}^{3}+6\, {q}^{4}+O\left({q}^{5}\right), {q}^{-7}-4\, {q}^{-5}+2\, {q}^{-3}+8\, {q}^{-1}-5\, q-4\, {q}^{3}+O\left({q}^{4}\right), {q}^{-8}-2\, {q}^{-7}-3\, {q}^{-6}+6\, {q}^{-5}+2\, {q}^{-4}-{q}^{-2}-10\, {q}^{-1}-2\, q+10\, {q}^{2}+O\left({q}^{3}\right), {q}^{-9}-4\, {q}^{-8}+2\, {q}^{-7}+8\, {q}^{-6}-5\, {q}^{-5}-4\, {q}^{-4}-10\, {q}^{-3}+8\, {q}^{-2}+9\, {q}^{-1}+14\, q+O\left({q}^{2}\right)\right] \]

_{List(ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)))}

_{Time: 0 sec}

\[ \left[11\, {q}^{-14}+11\, 17\, {q}^{-13}+11\, 101\, {q}^{-12}+11\, 336\, {q}^{-11}+{11}^{2}\, 84\, {q}^{-10}+11\, 2026\, {q}^{-9}+11\, 4036\, {q}^{-8}+11\, 7201\, {q}^{-7}+11\, 12328\, {q}^{-6}+11\, 19500\, {q}^{-5}+11\, 29765\, {q}^{-4}+O\left({q}^{-3}\right), {q}^{-5}-7\, {q}^{-4}+17\, {q}^{-3}-14\, {q}^{-2}+2\, {q}^{-1}-21+36\, q+13\, {q}^{2}-26\, {q}^{3}-24\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), {q}^{-5}+4\, {q}^{-4}+6\, {q}^{-3}+8\, {q}^{-2}+13\, {q}^{-1}+12+14\, q+24\, {q}^{2}+18\, {q}^{3}+20\, {q}^{4}+32\, {q}^{5}+O\left({q}^{6}\right), {q}^{-6}+2\, {q}^{-5}-{q}^{-4}-2\, {q}^{-3}-{q}^{-2}-6\, {q}^{-1}+4\, q-5\, {q}^{2}+4\, {q}^{3}+6\, {q}^{4}+O\left({q}^{5}\right), {q}^{-5}+{q}^{-4}+{q}^{-3}+2\, {q}^{-2}+2\, {q}^{-1}+3+4\, q+5\, {q}^{2}+6\, {q}^{3}+8\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), {q}^{-7}-4\, {q}^{-5}+2\, {q}^{-3}+8\, {q}^{-1}-5\, q-4\, {q}^{3}+O\left({q}^{4}\right), {q}^{-8}-2\, {q}^{-7}-3\, {q}^{-6}+6\, {q}^{-5}+2\, {q}^{-4}-{q}^{-2}-10\, {q}^{-1}-2\, q+10\, {q}^{2}+O\left({q}^{3}\right), {q}^{-9}-4\, {q}^{-8}+2\, {q}^{-7}+8\, {q}^{-6}-5\, {q}^{-5}-4\, {q}^{-4}-10\, {q}^{-3}+8\, {q}^{-2}+9\, {q}^{-1}+14\, q+O\left({q}^{2}\right)\right] \]

_{List(ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)))}

_{Time: 0 sec}

\[ O\left({q}^{6}\right) \]

_{ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

\[ O\left({q}^{6}\right) \]

_{ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

-------------------------------------------------------------------
--endtest
-------------------------------------------------------------------

Checking modularity

-------------------------------------------------------------------
--test:modularity
-------------------------------------------------------------------

We are looking for an expression of $S=\sum_{k=0}^{\infty}p(11k+6)q^k$ in terms of Dedekind eta-functions.

Note that the partition series is (up to a factor of $q^{1/24}$) given by $\eta(\tau)^{-1}$. Thus for the divisors $\{1,11\}$ of $M=11$, that gives us the vector $s=(-1, 0)$. The cofactor exponent from the $F$ given at the beginning of the previous section is $r=(10,2,11,-22)$ and corresponds to the divisors $\{1,2,11,22\}$ of $N=22$.

The following function, checks whether all the data corresponding to $F$ from above indeed leads to a modular function for $\Gamma_0(22)$.

sspec := eqSPEC(22, [10,2,11,-22])
rspec := eqSPEC(1, [-1])
assertTrue(modular?(sspec,rspec,11,6)$QMOD0)

\[ \left[\left[1, 10\right], \left[2, 2\right], \left[11, 11\right], \left[22, -22\right]\right] \]

_{QEtaSpecification}

_{Time: 0 sec}

\[ \left[\left[1, -1\right]\right] \]

_{QEtaSpecification}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

-------------------------------------------------------------------
--endtest
-------------------------------------------------------------------

Express in Klein's J-invariant

-----------------------------------------------------------------
--NONWORKING test:Klein
-----------------------------------------------------------------

The test is currently disabled, because for some reason it takes too long in newer versions (2.0) of QEta.

Let $$ U(v):=t^v q \prod_{n=1}^{\infty}(1-q^{11n})\sum_{k=0}^{\infty}p(11k+6)q^k. $$

Below, we show that $U(7)=11 t^2 h(t, j, j_p)$ where $j$, $j_p$, and $t$ are Laurent series with integer coefficients and $h$ is a polynomial with rational coefficients having no denominator that is divisible by 11. In other words, we can give a relation $z U(7) = 11 t^2 k(t, j, j_p)$ with an integer polynomial $k$ and an integer $z$ such that $\gcd(z,11)=1$.

We choose Klein's $j$-invariant (https://en.wikipedia.org/wiki/J-invariant#The_q-expansion_and_moonshine) expressed as a Laurent series in terms of $q = \exp(2\pi i \tau)$ as the series $j$ above.

The series $j_p$ is given by replacing $q$ by $q^{11}$ in $j$.

The series $t$ is given by $$ t = \left(\frac{\eta(\tau)}{\eta(11\tau)}\right)^{12} $$ (again in terms of $q = \exp(2\pi i \tau)$) where $\eta$ is Dedekind's eta-function (https://en.wikipedia.org/wiki/Dedekind_eta_function#Definition).

Construction of Klein's $j$-invariant

Klein's $j$-invariant can be given by the expression $$j(\tau) = E_4^3/\Delta$$ where $$\Delta=\frac{E_4^3-E_6^2}{1728}=\eta(\tau)^{24}$$ and the normalize Eisenstein series $E_4$ and $E_6$ are given by $$E_4=1+240 \sum_{n=1}^\infty \sigma_3(n)q^n$$ and $$E_6=1-504 \sum_{n=1}^\infty \sigma_5(n)q^n$$ where $q = \exp(2\pi i\tau)$ and $\sigma_k(n)$ denotes the sum of the $k$-th power of the positive divisors of $n$.

All the above are series with integers coefficients. For our computation, however, we compute with integers localized at the prime 11, i.e. these are rational numbers with denominators not divisible by 11.

The following function simply maps the integer coefficients of the series into the localized coefficient domain and eventually creates an object that attaches the indeterminate $J$ to the series.

i := kleinJInvariant()$QFunctions(C, L); --: L
j := toX1(C, i, J); --: X
x1A1(C,j)
x1Pol(C,j)

ip := multiplyExponents(i, 11)$L; --: L
jp := toX1(C, ip, Jp); --: X
x1A1(C,jp)
x1Pol(C,jp)

The series $t$

The value $t$ is given by $$t = \left(\frac{\eta(\tau)}{\eta(11\tau)}\right)^{12}$$ in terms of $q = \exp(2\pi i \tau)$.

tt := specM0A1(C)(eqSPEC([[1,12],[11,-12]])); --: L
t := toX1(C, tt, T); --: X
x1A1(C,t)
x1Pol(C,t)

Construct elements suitable for the samba algorithm

m1 := t; --: X

We need some auxiliary values $m_2$ and $m_3$.

m2 := j*t^3; --: X
m3 := jp*t;  --: X

We want yet another value to end up in an algebra basis of orders -2 and -3.

We ignore this for now, because with the data that produces a relation among $F$, $J$, $J_p$, and $T$, the computation did not finish.

m := jp; mp := j;
while (d := qetaGrade(m)$X) > 0 repeat (_
    c := qetaLeadingCoefficient(m)$X;_
    e := d::NN;_
    m := m-c*j^e;_
    mp := mp-c*jp^e)

mp

[-q^(-121)+O(q^(-110)),_
-Jp^11_
+11*744*Jp^10_
+11(-2570796)*Jp^9_
+11*4880620256*Jp^8_
+11*(-5550817150590)*Jp^7_
+11^2*351821430873072*Jp^6_
+11^2*(-147929969197380488)*Jp^5_
+11^2*35519316022933293120*Jp^4_
+11^2*(-4373015221856525734395)*Jp^3_
+11^2*224874476512864841450600*Jp^2_
+11^2*(-3096214928567780111700972)*Jp_
+J]

mgens := [m1,m2,m3]

The algebra basis computation

Then we compute the algebra basis for the elements $t=m_1$, $m_2$, and $m_3$.

Since we compute in an extended algebra. The second component of each entry in the basis gives its expression in terms of the original generators.

--approx. 75 seconds
ab := samba(mgens)$QSAMBA(C,X1,QTOPRED);

assertEquals(# basis ab, 12)

Show the powers of 11 in the coefficients of the representation (in terms of J, Jp, T) of the algebra basis elements.

pows := [[exponent(x) for x in coefficients second ab.n] for n in 2..#ab]
powsExpected := [_
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,_
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0,_
   1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1,_
   1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 0, 0,_
   1, 1, 1, 0, 3, 3, 0, 0, 0, 3, 1, 3, 4, 4, 4, 5, 4, 6, 6, 6, 8, 8],_
  [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,_
   0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0,_
   0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 3, 2, 0, 1, 0, 1,_
   0, 2, 2, 3, 0, 1, 0, 1, 0, 3, 3, 1, 2, 3, 1, 3, 4, 1, 1, 1, 4, 4, 2, 2,_
   4, 4],_
  [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,_
   1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0,_
   0, 0, 0, 2, 2, 0, 0, 2, 0, 2, 3, 0, 0, 0, 3, 4, 3, 4, 3],_
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1,_
   0, 2, 1, 1, 1, 1, 1, 1, 2, 3, 0, 2, 0, 3, 3, 1, 3, 1, 3, 3, 1, 1, 3, 3,_
   2, 2, 3],_
  [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,_
   0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 1,_
   0, 1, 0, 2, 3, 0, 0, 3, 0, 3, 4, 1, 1, 1, 3, 3, 2, 2, 3],_
  [0, 0, 0],_
  [0],_
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 3, 2, 0, 2,_
   0, 2, 2, 0, 2, 0, 2, 2, 2, 2, 1, 1],_
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 1, 0,_
   1, 1, 0, 0, 1, 1],_
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],_
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]$List(List NN)
assertEquals(pows, powsExpected)

The (absolute value of the) order and the power of 11 in the leading coefficient of the elements in the algebra basis.

gl := [[qetaGrade i, unitCanonical qetaLeadingCoefficient i] for i in basis ab]
glExpected := [[7, 11^18], [8, 11^8], [9, 11^4], [11, 11^3],_
  [12, 11^3], [13, 11^3], [14, 1], [16, 1], [17, 11^2], [18, 11],_
  [11*2, 1], [23, 1]]$List(List C)
assertEquals(gl, glExpected)

We define $$ U(v):=t^v q \prod_{n=1}^{\infty}(1-q^{11n})\sum_{k=0}^{\infty}p(11k+6)q^k. $$

Let's do this step by step. We take the partition series.

ps := partitionSeries(1)$QFunctions(ZZ, LZZ) --: LZZ

We also take the sub-series given by the (11n+6)-th term of ps as defined earlier. The coefficients of the series r11 we want to check for divisibility by 11.

r11 := choose(11, 6, ps) --: LZZ

The following function simply maps the integer coefficients of the series into the localized coefficient domain or into the rational number domain Q.

abfmap(A, B, f, x) ==> map(f, x)$QEtaLaurentSeriesFunctions2(A, B)
abffmap(A, B, f, x) ==> abfmap(A, B, f, x::L1(A))::A1(B)
abmap(A, B, x) ==> abffmap(A, B, (c: A): B +-> c::B, x)
zcmap(x) ==> abmap(ZZ, C, x)
cqmap(x) ==> abmap(C, QQ, x)
zqmap(x) ==> abmap(ZZ, QQ, x)

Putting all together (except the factor $t^v$) gives the following expression.

q := monomial(1, 1)$LZZ
fz := q*eulerFunction(11)$QFunctions(ZZ, LZZ)*r11; --: LZZ
f := embed(zcmap fz, F)$X

Now let's check what the smallest $v>0$ is such that $U(v)=t^v f$ is in the $\mathbb{Q}[t]$-module generated by $\{1, z_1, \ldots, z_{12}\}$ where the $z_i$ are given by the list ab.

Although for the coefficient domain $\mathbb{Q}$ we find $v=3$, with the coefficients being in $C=\mathbb{Z}_{(11)}$, the reduction to zero only starts to work with $v=7$.

qab := [cqmap first x for x in basis ab];
qt := cqmap first t
qreds := [zero? reduce(cqmap(first(t^v * f)), qt, qab)$QEtaTopReduction(QQ, LQQ)_
          for v in 1..7]
qredsExpected := [false, false, true, true, true, true, true]
assertEquals(qreds, qredsExpected)

With $v=3$ the reduction does not lead to 0 in case the coefficients are non-rational, but rather lie in $\mathbb{Z}_{(11)}$.

assertFalse(zero? reduce(t^3*f, t, ab)$QTOPRED(C,CA1))

assertEquals(_
  [zero? reduce(t^v * f, t, ab)$QEtaTopReduction(C, X) for v in 1..10],_
  [false, false, false, false, false, false, true, true, true, true])

Since all involved series in the algebra basis ab have their representation in terms of $J$, $Jp$, and $T$ attached, we have found a relation in $F$, $J$, $Jp$, and $T$.

result := reduce(t^7 * f, t, ab)$QEtaTopReduction(C, X);
assertTrue(zero? result)
res := second result;
assertEquals(variables res, [T, Jp, J, F])

The leading coefficient is a rational number whose numerator and denominator are not divisible by 11, i.e., the exponent (of 11) of the coefficient is zero.

cf := coefficient(res, 'F, 1);
lcf := leadingCoefficient cf
assertTrue(zero? exponent lcf)

Let's normalize the coefficient in front of the variable $F$ such that it becomes 1, i.e., we multiply by the inverse of the coefficient.

r := (inv(unitPart lcf)::C)*res;

Let us investigate the coefficients of all monomials of $r$.

mr := [(inv(unitPart leadingCoefficient x)::C)*x for x in monomials r];
sort(smaller11, mr)
sort(smallerT, mr)

The above shows that we have found a relation $$ U(7):=t^7 q \prod_{n=1}^{\infty}(1-q^{11n}) \sum_{k=0}^{\infty}p(11k+6)q^k = 11 t^2 h(t,j,j_p) $$ where $h$ is a polynomial in $\mathbb{Z}_{(11)}[t,j,j_p]$.

Let us create output that can directly pasted into a journal paper.

tups := [(d:=degree(x, [T,J,Jp]); [d.1-2,d.2,d.3]) for x in rest sort(smaller11, mr)];
assertEquals(#tups, 161)

tupsExpected: List List ZZ := [_
  [0, 0, 0], [0, 0, 1], [0, 0, 2],_
  [1, 0, 0], [1, 0, 1], [1, 0, 2], [1, 0, 3],_
  [2, 0, 0], [2, 0, 1], [2, 0, 2], [2, 0, 3], [2, 0, 4],_
  [2, 1, 0], [2, 1, 1],_
  [3, 0, 0], [3, 0, 1], [3, 0, 2], [3, 0, 3], [3, 0, 4],_
  [3, 1, 0], [3, 1, 1], [3, 1, 2],_
  [4, 0, 0], [4, 0, 1], [4, 0, 2], [4, 0, 3], [4, 0, 4],_
  [4, 1, 0], [4, 1, 1], [4, 1, 2], [4, 1, 3],_
  [4, 2, 0],_
  [5, 0, 0], [5, 0, 1], [5, 0, 2], [5, 0, 3], [5, 0, 4],_
  [5, 1, 0], [5, 1, 1], [5, 1, 2], [5, 1, 3],_
  [5, 2, 0], [5, 2, 1],_
  [6, 0, 0], [6, 0, 1], [6, 0, 2], [6, 0, 3], [6, 0, 4],_
  [6, 1, 0], [6, 1, 1], [6, 1, 2], [6, 1, 3],_
  [6, 2, 0], [6, 2, 1], [6, 2, 2],_
  [7, 0, 0], [7, 0, 1], [7, 0, 2], [7, 0, 3], [7, 0, 4],_
  [7, 1, 0], [7, 1, 1], [7, 1, 2], [7, 1, 3],_
  [7, 2, 0], [7, 2, 1], [7, 2, 2],_
  [7, 3, 0],_
  [8, 0, 0], [8, 0, 1], [8, 0, 2], [8, 0, 3], [8, 0, 4],_
  [8, 1, 0], [8, 1, 1], [8, 1, 2], [8, 1, 3],_
  [8, 2, 0], [8, 2, 1], [8, 2, 2],_
  [8, 3, 0], [8, 3, 1],_
  [9, 0, 0], [9, 0, 1], [9, 0, 2], [9, 0, 3],_
  [9, 1, 0], [9, 1, 1], [9, 1, 2], [9, 1, 3],_
  [9, 2, 0], [9, 2, 1], [9, 2, 2],_
  [9, 3, 0], [9, 3, 1],_
  [10, 0, 0], [10, 0, 1], [10, 0, 2], [10, 0, 3],_
  [10, 1, 0], [10, 1, 1], [10, 1, 2], [10, 1, 3],_
  [10, 2, 0], [10, 2, 1], [10, 2, 2],_
  [10, 3, 0], [10, 3, 1],_
  [10, 4, 0],_
  [11, 0, 0], [11, 0, 1], [11, 0, 2],_
  [11, 1, 0], [11, 1, 1], [11, 1, 2],_
  [11, 2, 0], [11, 2, 1], [11, 2, 2],_
  [11, 3, 0], [11, 3, 1],_
  [11, 4, 0],_
  [12, 0, 0], [12, 0, 1], [12, 0, 2],_
  [12, 1, 0], [12, 1, 1], [12, 1, 2],_
  [12, 2, 0], [12, 2, 1], [12, 2, 2],_
  [12, 3, 0], [12, 3, 1],_
  [12, 4, 0],_
  [13, 0, 0], [13, 0, 1],_
  [13, 1, 0], [13, 1, 1],_
  [13, 2, 0], [13, 2, 1],_
  [13, 3, 0], [13, 3, 1],_
  [13, 4, 0],_
  [14, 0, 0], [14, 0, 1],_
  [14, 1, 0], [14, 1, 1],_
  [14, 2, 0], [14, 2, 1],_
  [14, 3, 0], [14, 3, 1],_
  [14, 4, 0],_
  [15, 0, 0], [15, 1, 0], [15, 2, 0], [15, 3, 0], [15, 4, 0],_
  [16, 0, 0], [16, 1, 0], [16, 2, 0], [16, 3, 0], [16, 4, 0]];

assertEquals(sort tups, tupsExpected)

Size of numerator and denominators of the coefficients of the linear combination of the powerproducts.

sz(x) ==> floor((log(abs(numer(x::QQ)::Float))/log(10.0))::Float)::ZZ
sizes := remove(0, sort [sz x for x in coefficients r])
assertEquals(min sizes, 2324)
assertEquals(max sizes, 2395)
degs := sort [degree(x, T) for x in mr]
degsExpected: List NN := _
  [2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6,_
   6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8,_
   8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10,_
   10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11,_
   11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12,_
   12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14,_
   14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15,_
   16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18]
assertEquals(degs, degsExpected)

Check the relation

Plugging in the series $f$, $j$, $j_p$, $t$, for the indeterminates $F$, $J$, $J_p$, $T$ in $r$ should result in the zero series.

-- approx. 45 seconds
fjjpt: List Symbol := [F, J, Jp, T];
z := eval(r, c+->c::A1, fjjpt, [first f, first j, first jp, first t])_
     $ PolynomialEvaluation(C, A1)
assertTrue(zero? z)

Let's check whether the relation interpreted with rational coefficients also yields zero. This checks against bugs in IntegerLocalizedAtPrime.

-- approx. 30 seconds
rq := r::Polynomial(Fraction Integer);
zq := eval(rq, c+->c::LQQ, fjjpt, [cqmap first f, cqmap i, cqmap ip, zqmap tz])_
      $ PolynomialEvaluation(QQ, LQQ)
assertTrue(zero? zq)

-------------------------------------------------------------------
--endtest
-------------------------------------------------------------------

The same algorithm can be used over the rational numbers $\mathbb{Q}$

-------------------------------------------------------------------
--test:Klein_rational
-------------------------------------------------------------------

However, that computation leads to only 5 elements in the algebra basis.

nn := 11
idxs := etaFunctionIndices nn

\[ 11 \]

_{PositiveInteger}

_{Time: 0 sec}

\[ \left[\left[1\right], \left[11\right]\right] \]

_{List(List(Integer))}

_{Time: 0 sec}

LQQ ==> A1 QQ
XQ ==> X1 QQ
i0 := kleinJInvariant()$QFunctions(QQ, LQQ); --: LQQ
i1 := multiplyExponents(i0, 11); --: LQQ
xj0 := toX1(QQ, i0, J0); --: XQ
xj1 := toX1(QQ, i1, J1); --: XQ
tt := specM0A1(QQ) eqSPEC(nn,[12,-12])
xt := toX1(QQ, tt,  T); --: XQ
xgens := [xt, xj0*xt^3, xj1*xt]; --: List XQ
xab := samba(xgens)$QSAMBA(QQ,X1,QRED);
assertEquals(# basis xab, 4)

_Void

_{Time: 0 sec}

_Void

_{Time: 0 sec}

_{ModularFunctionQSeriesInfinity(Fraction(Integer))}

_{Time: 0 sec}

_{ModularFunctionQSeriesInfinity(Fraction(Integer))}

_{Time: 0 sec}

_{QEtaExtendedAlgebra(Fraction(Integer),ModularFunctionQSeriesInfinity(Fraction(Integer)),QEtaLazyAlgebra(Fraction(Integer),Polynomial(Fraction(Integer))))}

_{Time: 0 sec}

_{QEtaExtendedAlgebra(Fraction(Integer),ModularFunctionQSeriesInfinity(Fraction(Integer)),QEtaLazyAlgebra(Fraction(Integer),Polynomial(Fraction(Integer))))}

_{Time: 0 sec}

\[ {q}^{-5}-12\, {q}^{-4}+54\, {q}^{-3}-88\, {q}^{-2}-99\, {q}^{-1}+540-418\, q-648\, {q}^{2}+594\, {q}^{3}+836\, {q}^{4}+1056\, {q}^{5}+O\left({q}^{6}\right) \]

_{ModularFunctionQSeriesInfinity(Fraction(Integer))}

_{Time: 0 sec}

_{QEtaExtendedAlgebra(Fraction(Integer),ModularFunctionQSeriesInfinity(Fraction(Integer)),QEtaLazyAlgebra(Fraction(Integer),Polynomial(Fraction(Integer))))}

_{Time: 0 sec}

_{List(QEtaExtendedAlgebra(Fraction(Integer),ModularFunctionQSeriesInfinity(Fraction(Integer)),QEtaLazyAlgebra(Fraction(Integer),Polynomial(Fraction(Integer)))))}

_{Time: 0 sec}

-- numOfGaps:=[384, -1]
-- numOfGaps:=[291, -1]
-- numOfGaps:=[197, -1]
-- numOfGaps:=[106, -1]
-- numOfGaps:=[15, -1]
-- numOfGaps:=[14, -1]
-- numOfGaps:=[13, -1]
-- numOfGaps:=[12, -1]
-- numOfGaps:=[11, -1]
-- numOfGaps:=[10, -1]
-- numOfGaps:=[9, -1]
-- numOfGaps:=[8, -1]
-- numOfGaps:=[7, -1]
-- numOfGaps:=[6, -1]
-- numOfGaps:=[5, -1]
-- numOfGaps:=[5, -1]
-- numOfGaps:=[5, -1]
-- numOfGaps:=[5, -1]
-- numOfGaps:=[5, -1]
-- numOfGaps:=[5, -1]
-- numOfGaps:=[5, -1]
-- numOfGaps:=[5, -1]
-- numOfGaps:=[5, -1]
-- numOfGaps:=[5, -1]
-- numOfGaps:=[5, -1]
-- numOfGaps:=[5, -1]

_{QEtaAlgebraBasis(QEtaExtendedAlgebra(Fraction(Integer),ModularFunctionQSeriesInfinity(Fraction(Integer)),QEtaLazyAlgebra(Fraction(Integer),Polynomial(Fraction(Integer)))))}

_{Time: 0.21 (EV) = 0.21 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

bas := sort(smallerGrade?, basis xab);
degs := [[qetaGrade i, unitCanonical qetaLeadingCoefficient i] for i in bas]
degsExpected := [[7, 1], [8, 1], [9, 1], [11, 1]]$List(List QQ)
assertEquals(degs, degsExpected)

Compiling function smallerGrade? with type (QEtaExtendedAlgebra(Fraction(
Integer),ModularFunctionQSeriesInfinity(Fraction(Integer)),QEtaLazyAlgebra(
Fraction(Integer),Polynomial(Fraction(Integer)))), QEtaExtendedAlgebra(
Fraction(Integer),ModularFunctionQSeriesInfinity(Fraction(Integer)),
QEtaLazyAlgebra(Fraction(Integer),Polynomial(Fraction(Integer))))) -> Boolean
 

_{List(QEtaExtendedAlgebra(Fraction(Integer),ModularFunctionQSeriesInfinity(Fraction(Integer)),QEtaLazyAlgebra(Fraction(Integer),Polynomial(Fraction(Integer)))))}

_{Time: 0 sec}

\[ \left[\left[7, 1\right], \left[8, 1\right], \left[9, 1\right], \left[11, 1\right]\right] \]

_{List(List(Fraction(Integer)))}

_{Time: 0 sec}

\[ \left[\left[7, 1\right], \left[8, 1\right], \left[9, 1\right], \left[11, 1\right]\right] \]

_{List(List(Fraction(Integer)))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

rspec := eqSPEC(1,[-1]); m := 11; t := 6;
sspec := cofactInfM0(nn, idxs, rspec, m, t)
xf := toX1(QQ, specM0A1(QQ)(sspec,rspec,m,t), F) --: X

_{PositiveInteger}

_{Time: 0 sec}

-- == z:=[zinhom=[[0]], zhom=[], zfree=[]]

\[ \left[\left[1, 12\right], \left[11, -11\right]\right] \]

_{QEtaSpecification}

_{Time: 0.04 (EV) = 0.04 sec}

\[ \left[11\, {q}^{-4}+165\, {q}^{-3}+748\, {q}^{-2}+1639\, {q}^{-1}+3553+4136\, q+6347\, {q}^{2}+3586\, {q}^{3}+7414\, {q}^{4}-4444\, {q}^{5}+583\, {q}^{6}+O\left({q}^{7}\right), F\right] \]

_{QEtaExtendedAlgebra(Fraction(Integer),ModularFunctionQSeriesInfinity(Fraction(Integer)),QEtaLazyAlgebra(Fraction(Integer),Polynomial(Fraction(Integer))))}

_{Time: 0 sec}

As we have seen above, $U(3)$ lies in the $\mathbb{Q}[t]$-module generated by 1 and the elements of the algebra basis.

qreds := [zero? reduce(xt^v * xf, xab)$QTOPRED(QQ,X1) for v in 1..7]
qredsExpected := [false, true, true, true, true, true, true]
assertEquals(qreds, qredsExpected)

\[ \left[\texttt{false}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}\right] \]

_{List(Boolean)}

_{Time: 0.13 (EV) = 0.13 sec}

\[ \left[\texttt{false}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}\right] \]

_{List(Boolean)}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

xr := reduce(xt^2 * xf, xab)$QTOPRED(QQ,X1);
assertTrue(zero? xr)
pol := x1Pol(QQ,xr);
cfq := coefficient(pol, 'F, 1)
lcfq := leadingCoefficient cfq
rq := inv(lcfq)*pol - F*T^2
fl := factors factor rq;
assertEquals(fl.1.factor, T)
assertEquals(fl.1.exponent, 2)
cfs := coefficients fl.2.factor

_{QEtaExtendedAlgebra(Fraction(Integer),ModularFunctionQSeriesInfinity(Fraction(Integer)),QEtaLazyAlgebra(Fraction(Integer),Polynomial(Fraction(Integer))))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

_{Polynomial(Fraction(Integer))}

_{Time: 0.04 (IN) = 0.04 sec}

\[ -\frac{377}{15797171629998}\, {T}^{2} \]

_{Polynomial(Fraction(Integer))}

_{Time: 0 sec}

\[ -\frac{377}{15797171629998} \]

_{Fraction(Integer)}

_{Time: 0 sec}

\[ \left(\frac{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}{14291452238531994355130293720977422217461910181149972915321956889133110260852886590459673387035400054802019248521925674160715596624103190749900911411200000000}\, {J0}^{2}-\frac{189753762252959744196104033897917525472504226759523418410425040837250517640142178784735342565965128076817}{9924619610091662746618259528456543206570770959131925635640247839675771014481171243374773185441250038056957811473559495944941386544516104687431188480000000}\, J0-\frac{576291199618072743284408400029034433233986765963800420102156550006016977633017050240898859367260745143059}{68920969514525435740404580058725994490074798327305039136390609997748409822785911412324813787786458597617762579677496499617648517670250726996049920000000}\right)\, {T}^{17}+\left(\frac{2203602608731576910218467952157266108416230074723912152858078409704735837054408210872221510179121025649406529}{155341872157956460381851018706276328450672936751630140383934314012316415878835723809344275946036957117413252701325279066964299963305469464672835993600000000}\, {J0}^{2}+\frac{44576738352283432703803818392006782510948791251795487437386631955925960885374361516337072165369816565850010621}{5050182989055395250670801714052790151928949195568860999206313779602977312477697559217775893263408323863707414753490482365632702292141311157862400000000}\, J0-\frac{11450194746555084574915392489904986188854280305564046961236523852276288933960224907708189792403433886585936551321}{310454817632997458290110720985252227432769361834707383497254999989857701904441042397859521566605669358638570178727461709989407737253381653135360000000}\right)\, {T}^{16}+\left(\left(\frac{1861103109538848055747192188998557612380226693790297185746321311866259254689351848540731969071372081227}{19577331833605471719356566741064961941728644083767086185372543683743986658702584370492703269911506924386327737701268046795500817293292042123151933440000000}\, J0-\frac{4348100583310137217102014713272799410661153233376440418369467831063933091690712893361309210592136555027}{79396956880733301972946076227652345652566167673055405085121982717406168115849369946998185483530000304455662491788475967559531092356128837499449507840000}\right)\, J1-\frac{211439618536437516325288092953470633596080158422852221398610550040809221875234888924611624421180740020793310308765817}{71457261192659971775651468604887111087309550905749864576609784445665551304264432952298366935177000274010096242609628370803577983120515953749504557056000000000}\, {J0}^{2}+\frac{543814075900373806024769785528088946350645239273014530313366608416974074748472157894385846033801000302729397631119611}{76343227769935867281678919449665716973621315070245581812617291074429007803701317256729024503394231061976598549796611507268779896496277728364855296000000}\, J0+\frac{1390862548762209163846490152114182930761982001984370346194710903195686724568281532811280109278454863112932355200436477501}{68920969514525435740404580058725994490074798327305039136390609997748409822785911412324813787786458597617762579677496499617648517670250726996049920000000}\right)\, {T}^{15}+\left(\left(-\frac{372268131413316544566585754444244861744583903451713957267785348870009019518613115646786504470015679655711520448911}{1786431529816499294391286715122177777182738772643746614415244611141638782606610823807459173379425006850252406065240709270089449578012898843737613926400000000}\, J0-\frac{30403047277783607066775855157585255427906838701430486266848084338447240028930634417120080712067524229482440075808773}{24811549025229156866545648821141358016426927397829814089100619599189427536202928108436932963603125095142394528683898739862353466361290261718577971200000000}\right)\, J1+\frac{2044650096176766610695166265588908482351815176626967687008804179304959776079814925049666817844581600640315651431009499451}{35728630596329985887825734302443555543654775452874932288304892222832775652132216476149183467588500137005048121304814185401788991560257976874752278528000000000}\, {J0}^{2}-\frac{1745025138235175729967251079625293585520416003709598151968578067206279599100166560346814850385904325476500719757883410861}{41770284554257839842669442459833936054590786865033357052357945453180854438052067522621099265325126422798643987683331211889483950103182258785484800000000}\, J0+\frac{5082434529329283626173333744482245679654562303275583230296588291239528246918565816919403381197033715773254434853719361117893}{3132771341569337988200208184487545204094309014877501778926845908988564082853905064196582444899384481709898299076249840891711296257738669408911360000000}\right)\, {T}^{14}+\left(-\frac{1412568970148406436681581022041490980323931124165036943504015798551426782063094715172266162601033007103047}{14291452238531994355130293720977422217461910181149972915321956889133110260852886590459673387035400054802019248521925674160715596624103190749900911411200000000}\, {J1}^{2}+\left(-\frac{977708900529262072323387378054033175694187888483962168151377452575115305288117044406341415816143216378483858826068771773}{3248057326939089626165975845676686867604979586624993844391353838439343241102928770559016678871681830636822556482255835036526271960023452443159298048000000000}\, J0+\frac{5480341108301319413087754682347573772259156115584195462953816166495622365412154296561085899918759500298155682855729400097}{381716138849679336408394597248328584868106575351227909063086455372145039018506586283645122516971155309882992748983057536343899482481388641824276480000000}\right)\, J1-\frac{571954277075025612895651032570339402614292015175656252120137950089355096640077053834314203170210524878434596716682112584719}{3248057326939089626165975845676686867604979586624993844391353838439343241102928770559016678871681830636822556482255835036526271960023452443159298048000000000}\, {J0}^{2}-\frac{121499337496090465270879044006741016428920981269772512224396500707530393392075917965149682029140122379106935428075646928909089}{69402934336305333892435381317877924521473922791132347102379355522208188912455742960662749548540210056342362317996919552062527178632979753058959360000000}\, J0-\frac{1557659800940077570529035717860120278518449421515514875233255308420501253976537592010747135413039650439746010727877327406492163}{160654940593299384010267086383976677133041487942435988662915174819926363223277182779311920251250486241533246106474350814959553654243008687636480000000}\right)\, {T}^{13}+\left(-\frac{17758677464292680754010096855311288147726175445574405864872955560722865976789434957844882792215945734636938757856293}{714572611926599717756514686048871110873095509057498645766097844456655513042644329522983669351770002740100962426096283708035779831205159537495045570560000000}\, {J1}^{2}+\left(\frac{32548590488294131680069885659463370063910328026380075181251603607881532198301220806476810024032053412967205766732996411873799}{1624028663469544813082987922838343433802489793312496922195676919219671620551464385279508339435840915318411278241127917518263135980011726221579649024000000000}\, J0-\frac{13164713802594498783817274650128872029384957033286072136879059290102227708930793490442338782011497939895248006645908014710503}{19977092612905923403015820306877099852195593718059431633736408694999539079068380119514438779068538723947177559326810579599318410918913254201753600000000}\right)\, J1+\frac{434239901904851079202897055343907622179812919960181588008669861453785721734638003792170362907030908638971799974738405555097767}{1624028663469544813082987922838343433802489793312496922195676919219671620551464385279508339435840915318411278241127917518263135980011726221579649024000000000}\, {J0}^{2}-\frac{35092666241515514381693343830489080665123740754910368221410151758878802048888706593440451066096120125529380076391935392291180427}{68351374725149192469822724025182804452966742142781856994767547105205034534994292309743616979622934146397781070754541983091882827441570968921702400000000}\, J0+\frac{18262291742359215208673930585203083003746606636829895984289574262267832578570887061401299448050173511635468561319642682181943}{2615435570323733231313221632380959110982800160358441660114531469742105811277420092483294340962634429016087229077492706127070603762844303360000000}\right)\, {T}^{12}+\left(-\frac{1966275739335255208007417210720955347574393159275976505493636835102278016126384975741336125711121050362414301768647744975177}{71457261192659971775651468604887111087309550905749864576609784445665551304264432952298366935177000274010096242609628370803577983120515953749504557056000000000}\, {J1}^{2}+\left(-\frac{3753926049007967543789697769960709958598078494454089027646575126994965424264259483043167888156130684087461500581869388840858401}{13421724491483841430437916717672259783491651184400801009881627431567534054144333762640564788725957977838109737529982789407133355206708481170079744000000000}\, J0-\frac{122813975965574220250185122845719328972597924051589251698723737970455200280473774966083336068356039914472811998864862958851595079}{665590477505979662438351520921235544669758631824285804817936277893888472569322731096155401075799747706857877504765575477495901451580450988359680000000}\right)\, J1-\frac{58631064168557507630436786390399463687112022761295969029522209764781041032133760382641028431464651381565200536277066533640168559}{295277938812644511469634167788789715236816326056817622217395803494485749191175342778092425351971075512438414225659621366956933814547586585741754368000000000}\, {J0}^{2}+\frac{516742120670190388524347795548463355310886328664357456439326496834256330747457368826219557099667769532699518733576925840843187433401}{102527062087723788704734086037774206679450113214172785492151320657807551802491438464615425469434401219596671606131812974637824241162356453382553600000000}\, J0+\frac{4154242982909966162540876440062989839836061917495678726517434147612067534623201070528855785476400200921184742757021581029159464085629}{113918957875248654116371206708638007421611236904636428324612578508675057558323820516239361632704890243996301784590903305153138045735951614869504000000}\right)\, {T}^{11}+\left(\frac{363408959434640751905489181678235931553808903641117205843751861950465858515837818735889338368211713578946894187408602493664698051}{295277938812644511469634167788789715236816326056817622217395803494485749191175342778092425351971075512438414225659621366956933814547586585741754368000000000}\, {J1}^{2}+\left(\frac{68972724435920474496126678693561864385297822899210051365045216713960423922104402844181342636975348907014195881238796288801}{91896458915985373844990156991702418606826042066970054931907492606278162289731352762830000533683844391416206254348725394797833933851459584000000000}\, J0+\frac{23405199435887417044161934334604896977393995191318292186100852708398326111802690571127516169886256638958659218280059153910885258591}{15267226876289745916869047135399330903052656274912186060926412129820199806788986443990086437262214462005311831752187175137789329336960234291200000000}\right)\, J1+\frac{37378415860986624843644530082406552767080856810675051356561560340890448474530983676511896721017866173736004942742740637065646717}{1011225817851522299553541670509553819304165500194580898004780148953718319147860762938672689561544779152186350087875415640263471967628721184047104000000000}\, {J0}^{2}-\frac{4864723216787659261808925489497945746318257443136696107957863989628410106564684991509333712276705874241976888249104864474572297493997}{4660321003987444941124276638080645758156823327916944795097787302627615991022338112027973884974290964527121436642355135210810192780107111517388800000000}\, J0-\frac{6603327126117480656681438876779573225271623196984503020550619287767476163421655858287693360930057346629131253276133374619393}{73568927185724705224462323858401512048585996060293884917689877191999212532928249163827428143564274499528440130019008033931988566016000000}\right)\, {T}^{10}+\left(-\frac{261436504833287894937283558220835960737693128056984274574654391618651417051257415814048238277250026346490146944311477948701106519}{5410894775845128575060639676545962420274803944527635964475560343304790991390580029284646155503309001345740672255586691960142453217782092791808000000000}\, {J1}^{2}+\left(-\frac{3825895216371138649778533378836853100090560574418959241516178143233096588160266866717774662820286019283761593756854756529864794559}{3824942858787073647887693564454904469504602788372984043853413346129248804258858296563284351304063259571989095904811282247686906584983893180416000000000}\, J0+\frac{9535627581147483767225817350504181804909213626106561037770719087013739166535197014148396330487682691719481014504659942427354737852079}{7002736294496536350299438975327792273714234902955589474226577464504306522948667335879750390645065311085081046795424696034275270894225562009600000000}\right)\, J1-\frac{781426434268609798329884148432847958452930549745209252955772475295490346145584111971997836464635631090282793154640050481173713497}{443693371619300543154972453476768918462533923451266149086995948150992861294027562401340984751271338110350735124958108740731681163858131608928256000000000}\, {J0}^{2}-\frac{214708308022651859133208375362873720083714829025800687365676757158644443066761775915301199252157836106410471732700029918012988133}{105447412382918718265005348993600973858721571801664506706117361750410241311684915782392651540550375676539680495641448673339009270389145600000000}\, J0+\frac{7691100831597693238723386649836506103466178785748489417142115067447096349476589772293135701794378485635134797275253700259650835820587}{64304281859472326449030660930466412063491596905010004354697681033097396904946440182550508637695732884160523845688013737688478153298673664000000}\right)\, {T}^{9}+\left(\frac{139680539099668773655298740954636697365133324778171772479718533358331876182450132106157953550091036500003945780980902980101183}{299585336383681496726725369513458852157370288874763967242991723382765149018475824834316965674812512151787298480516738517444766451367936000000000}\, {J1}^{2}+\left(\frac{10373341301241868499651636352606370985002375259461750311209150422509033165937842877904978326103488634714122453448265072268430569}{15653333826047358203971400557078225025222597593706417288446317546749479036215361847593061456508953759930886345606999587536489047083974656000000000}\, J0-\frac{6301995368999795000090604756192300427929987215913888036543698279119549513711304697360103545998591859575106055734495068157579680123}{530261985977214031299844192312948002209437587862683512481243819334331945671252095108166038499625804875707532032757438602184588315852800000000}\right)\, J1+\frac{679960706889992440879822297448736018628136806220927385178222278602516119867350144404250023783135009761873915082118970557947}{118378337059483146417533716712904076753245894302404780743875276447292935211378673972422527264848962809477327988652934380744698418572558336000000000}\, {J0}^{2}+\frac{294415215081267201367995180098016459404914163178360479961272705797323381294887547813097330568951542461019340837316938285011985659}{666193026257159559171516818756450785428558372494276139390807366310255342190587310878534148020675813355716382757710580033032666700840960000000}\, J0-\frac{2088147709184465513741841404917765789642502089441151445086389405564773575259783387236873042719954302570984683605873177609339024668347}{24156379361184194759215124316478742322874379002633360013034440658564010858357039888260897309427397777671120903714505536321742356611072000000}\right)\, {T}^{8}+\left(-\frac{867777149493819381800848764157153092515089521131381840312767904489467964064348892930417995270051146476906985232515350678912999}{777559099964476256786801724029194480020495251502016990317169967115980852585348087132302759076762206254211962451993447419090302926848000000000}\, {J1}^{2}+\left(-\frac{66820545757840831962157812973512265739539635153782017825701968722719788360534096051214539590308006994267413156941594523433253}{777559099964476256786801724029194480020495251502016990317169967115980852585348087132302759076762206254211962451993447419090302926848000000000}\, J0+\frac{216886002998564915043810264347437690965852330162904762170991688891223671948620764375756450548156356046824414105386694541760514777}{98176654035918719291262843943080111113698895391668811908733581706563238962796475648018025135954824021996459905554728209481098854400000000}\right)\, J1-\frac{3274724134487388027210860148426321705569393244378300110304188904758713454630287294980231140611403896953096897779618411}{684252007968739105972385517145691142418035821321774951479109571062063150275106316676426427987550741503706526957754233728799466575626240000000}\, {J0}^{2}+\frac{47160348379929765354137189084140094656364940183400545420356296488439010447071547983998888070674239309302799654023222266789421}{36892469373745854143611192910287867840862007274509087704834668902155751287262029668975717520032713374725377790596652525923021619200000000}\, J0-\frac{1306382925206108076058975366394952346942928255379000001233351184857766570766413426690107261674624282673552325405088942416509553337}{45452154646258666338547612936611162552638377496142968476265547086371869890183553540749085711090196306479842548867929726611619840000000}\right)\, {T}^{7}+\left(\frac{3493745337039841917429452226528017588167904232919510880758999510050705888937234169937228351631146660846991792479920048861}{2676291713451377464346214812086403170949151112789508147558778866924805824015353660375609623503881488829907505893808078323712000000000}\, {J1}^{2}+\left(\frac{4307860016466131268701459705319898143108409231915453996631728722909863539149642190915932290700580404680379555078632153}{1714499378929788688096793863992852031389299931630778657029842711623703731009835938678124915057174078781659495963220800176128000000000}\, J0+\frac{118362624181083324659310463923595398684520958986349157645278148039911433633302890593703609120259704104261635407613564118533}{9140797408532553276202896260937216141759983641089771417496370732236803684096357947668315636498820499660636089172633845760000000}\right)\, J1+\frac{40848885283602744395225920444274431166417083897023969303298233515382265034776538026686428501831209433}{34859868438513205209297948966670017999812550733749586759687133023264518183381167300201332968618713486487052637403139276800000000}\, {J0}^{2}-\frac{974843919029237139962196690756375300701016939108715787077454753758899120701967462349939316604788956132587803530122101981}{8596919962724866356268823933411451781325264614444930018155336673668713864892624649782050856127140679930828241866862131937280000000}\, J0+\frac{259815057421784212094589547477054420198676004583602597761298438823151351159666159017112511385280233029227414517359864770434073}{3618232307544135671830313103287648056113326857931367852758980081510401458288141687618708272780783114449001785297500897280000000}\right)\, {T}^{6}+\left(-\frac{4112458048644272163725122655576782604911760326118613919393623273903642754900749843201757074655153781229053886328155733697}{5450594420475823238015029142099957405240088946948225545940599365116244607466181523884980264073325395906946631397315444736000000000}\, {J1}^{2}+\left(-\frac{146613978669062331367487932312765670755518502171997593941414560920758329702798642030921885383604081708629277076059}{28687339055135911779026469168947144238105731299727502873371575605874971618243060652026211916175396820562877007354291814400000000}\, J0+\frac{12384550852910833184602613947515584688012557288143176276584558600311922619894823787545637406689204940082519564030115341127}{24578798793631959045883067920724916149170675265819920391146281408352473879266691575960408838714490421658309124266393600000000}\right)\, J1+\frac{550520332787030840408191468936699167761135219818459411762354928647396254889788795330602508930006599811462715179245587}{1261711671406440564355330819930545695657427996978755913412175778962093659135690167565967653720677174978459868379008204800000000}\, J0-\frac{10280503400151476921040997136629266206365572145065690282763543250310964809149979706176696980090195770791867386219756485692301}{477921087653954759225504098458540036233874241279831785383399916273520325430185669532563505197226202643356010749624320000000}\right)\, {T}^{5}+\left(\frac{856297070107132269023240327706636612378648481356294121728234482205368832261011789402842364805130330473571472321687}{12018431459590544454013272806484201568119357701494053006828704329111631209941273022874010071646659261386715037696000000000}\, {J1}^{2}+\left(-\frac{5087886970414485777298769214435063677168038953440981800904049382590914365692595554272197037320223}{1383241393006092402417840828991601941495577959495645677702732039205276107458483889967784713538246132367360000000}\, J0-\frac{26940885260785098220005600520903942020154849653560437283531425524104784284923465251723778764218083627763512095454643}{161131978677265553263124153597911199170361759027907531514378472654421065325172086782289392614669472249193431040000000}\right)\, J1+\frac{149279624869049639718092256604445093277463389431729413467333634981875066004409242287407703804032653}{61707935476882899952307010315569797723386061193055748844182990415657595238286809091340618053956202460610560000}\, J0-\frac{196259433355388201111353090428652970516889199897697426595589291599489623282588073444360850851212239094871649696689988681}{494585656773273434321533860349144097453471510349549506453856145230931325511986544151193830108916018987107614720000000}\right)\, {T}^{4}+\left(-\frac{12365648862625629663175034687378442996949519246184293912261308911137199944468933401916827438601379207290174001}{15101977019162782477675843082726406144139476165652667311106828621530746734192961161546703619951899315693158400000000}\, {J1}^{2}+\frac{80244392560255472197692211924862642814174625971133158660224067373356688357867751169290782023390818110340280746213}{402020221574935181697389341322577941337046240520846467772519743397230526488932993883766415808904727153868800000000}\, J1+\frac{4989617377280807444340788015249516257734572462991363153316407992138805657732658737441813833480429152936336939833937}{1675084256562229923739122255510741422237692668836860282385498930821793860370554141182360065870436363141120000000}\right)\, {T}^{3}+\left(\frac{1102730502447904230015254983629551598228648470099906674222592752835110442744173351189591962979541621107}{569986704885943782779720148029827602488298122634118353657025859465099738843140732764202215381191884800000000}\, {J1}^{2}+\frac{283621438555364650902947726943295091413610023693571965754686950633251870952011916238680939174811434598613}{395824100615238738041472325020713612839095918495915523372934624628541485307736619975140427348049920000000}\, J1-\frac{35430664594528061000407857237764185352654118242576244464958495982227667948861366180581943276915092930396449}{2748778476494713458621335590421622311382610545110524467867601559920426981303726527605141856583680000000}\right)\, {T}^{2} \]

_{Polynomial(Fraction(Integer))}

_{Time: 0.01 sec}

_{List(Record(factor: Polynomial(Fraction(Integer)),exponent: NonNegativeInteger))}

_{Time: 0.11 (EV) = 0.11 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

\[ \left[1, -\frac{65921760}{13019}, -\frac{28829882880}{13019}, \frac{202731440003305075740099051598468481974293166874599918062943213692835697009005555400244378936479134359745400668}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{126147176942266658807301980021719289746343852977961028353211181132491405597990281162744084475914059461900154856240448}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{527097348947337038914889184648886941818822831243910892689805240093778502682810276583446192248316283260386076328586808320}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{327770}{13019}, -\frac{188820000}{13019}, -\frac{211439618536437516325288092953470633596080158422852221398610550040809221875234888924611624421180740020793310308765817}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, \frac{101801995008549976487836903850858250756840788791908320074662229095657546792913987957829030377527547256670943236545591179200}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{288409258111331692215208197942396972522804587931479034986935252886657599206478858643747043459980400415097653174362507974607360}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{2978145051306532356532686035553958893956671227613711658142282790960072156148904925174292035760125437245692163591288}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{17512155232003357670462892570769107126474339092023960089704496578945610256664045424261166490150893956181885483665853248}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{4089300192353533221390332531177816964703630353253935374017608358609919552159629850099333635689163201280631302862018998902}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{597049880896335964953915153387316448524297213173216750107137174226225327154527387623300620170434849535831862260841261725625984}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{23185659728037845559912654675628446210929740862558948649954612057571428700182743771520964672750372048660324871447879427870938434560}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{340789}{13019}, -\frac{21509595811643765591114522317188729865272133546647167699330303956652536716338574976939511147955150760326644894173512979006}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, \frac{205183971094801398826005535307093162033382804967472278132990877273596101361031056863247056092958355691162948766118508739631680}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{12582994095650563483704322716547466857514424333864437546643034901965812126081695184354912469744631547325561127767006476863818}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{25019143577194948608579412741868110103043408463071555717247727425694658607296273027383622523440534000305700143349337215600959606880}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{138565428282538894075682463129398578069088253307501123582669795103498235070542604249040517653677349921614679364564503680123382200542720}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{355173549285853615080201937106225762954523508911488117297459111214457319535788699156897655844318914692738775157125860}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{1432137981484941793923074969016388282812054433160723307975070558746787416725253715484979641057410350170557053736251842122447156}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{9417930936665683673952607746504994018845763181956589718146183983664812885147418216714525625940769534217541261970429426459776162176}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{19106555683813447484927470435131935375911768478247989872381473903966571756324072166855495967909359980114759198888489844424301748}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{7337455399105995871039497874829300898109392706962699070678205810960450962798041884209277032507905564806687021412636987303378333120576}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{99790135594798299184757119341763444122874286366697084107424411544363708765318683135451557126323127199885998118356742799385345203418178560}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{1966275739335255208007417210720955347574393159275976505493636835102278016126384975741336125711121050362414301768647744975177}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{19985902284918419203136350927270819819576169904473569983190365976121195918782917487721825836543239762081645029097872626188730126924}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{2637042041696655165336734928123630116606987009567876158194724305903538787590288020267746818725969952077173903589706519074388033281075360}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{14188717528790916846565702306476670212281109508233624505144374763077011929776370012599128880414445634338778529779050101140920791278}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, \frac{72029717684459178637185887917091404023495067125182114567990599046720658455549577555424396503236889730702051315310754846806813582716632192}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{521161414086407402996008063859934251778985508523201884142046362633547186007577519260842439084213739126125205296148077000534936416041597811200}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{87944968183183061961128381966133095436021754681150363814187950592012737760832752134085219885107234686105148393352881803466856928342}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, \frac{53632120794695830912206227533065148958531099918206497766701158662650798676606239382856270042889635922426294688769460375084675329149784}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, \frac{21909302362616758097320403565590870166360285319199681585971576442825998893459218525390469521859928241122735096654304494485559155197427210816}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{2641308378400758857951297073743176644733001665669541829060066099928682651004257430517036669894006495300881053273973024377606859610088}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{14918276969958238796085534320306148376357479913301627565370176699330233514633820547402318986076840874891428132950022955914470797230231056128}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{1282758058984207389063585666629860027836905558460982387683925404792699667912339480336716007090944104743175171144675829335869415377823073139737600}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{3452578064272279598938645255471955889646829709269840701170858482815185208136808762089999191468730307436543941753321242479533476291300458}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{71475053057047437079515979186626267320231445044206745357428952712916678879896700141604661957736906508233614258591684985691385583308495044}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, \frac{19460673715144570548653187315390572723504760653582168116186521073738537080890411701534172718683623261853094342378497132851062128729049897967488}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{125849508665393876630826171989258596556802917967015695397780112918814015737092466817202223560466033022721134120347934770043507732405347}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{29099751809735313465617939359361506908545257867053435867346033871485651818033037929116829671211371659440698421907303363483585846895561573291616}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{1709326300178886760339902636722338237240103741114595584650406480772187227059352630554482063544847434316403041583038768722946831001529285539006809600}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{33316679936542660650014671785267740432756345791354748572139604693250398307863477120619059636625900344180460609929908187055420781653653831668}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, \frac{47354165383606433769836551028511717917009044977282665055123709863616780012994117313291159454949323079128604889410409366855465169223220922544}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{169849372961426211936662144056786093531356309062630777687954510162957343239765540549462274979401837548695289028757044280672713450703514692388212992}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{410447815368188279546887934658447945214143531484352041922472152400783767457663359817348586026531399938373902822659200515688476249696512}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, \frac{6315918688417575634719611145267828889724296194411423271887074292838904194591511866800340313080878841263185365381164597064405140872970601198092480}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{1235394709058205388074979662584023668475868047325940685713751594754609107637109940718015058504945039689228953909393411159151783314736057480444195251200}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{79748251202043352046988384736898273819904663235631103659551082412485662948700043317845678530521645610798055746462368370548774340177973354841678}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{6140772053818497906187359425928131122955307809551325838134558308514841779874866881564709312326661153078064887470250657999054116650912864939466}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, \frac{31571823092749503807812305516077714453622520259145346892253921914522875738640540487058348028134928061903075120026220178041187005276451661477951571296}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{68396676980643468757414525264135709037741270378809237752598779772907487938009480905153334838807982614465296681509714257637178155752805}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{18269036414893222224088492136848146816251892242769217116350862231413907649365163965625732567483861587469669711728055924744535636937552164612792256}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{410764007253803828724825846689932874732828181523100300592599627450931791441614887921687801616244895984912797681678279942634409373983781670233590809500160}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{93283356158338119305550891466366455908188104940470504149832570439861578910077553522879389613889316251948453913770213253861258696077582921576580968}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, \frac{179543884448757703857429386153383704323388343288756988287188632320269351854715373231651886515356980213808523043389106651483282859264566891577856}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, \frac{185057573722425008395667027976476897275973072577091775523796006478637006057415012960815913962170251835936948311157264681872311735545917832079539334384960}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{16746761223657104536760538428033843516856715104616580990771162303497048328673522113735273188585692254194293223657018801310856305722}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{1620572876011032312283832430146862073584888862037974385551652138795988905384750147972287421379732476753834909214351936570528674203846183221277338240}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{1026228878740833400953107591254701546539221843663211957309835877162135731738943658916474908271852934841759395881645160102791892773053174737548546977922769920}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{53914301130513546153642946968024711375393060247784300680558884917479251912762153052461971509376928423043150536139555057422214620340015755937794821587}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{73040119532275870521047874588342446689596701653883928831260756868488316084555212144680348984779857733928332249668766914495432218059956491599782}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{7201052358014438167079286576980308190464302805419687655058844973806285051142791838474542339396525822669344698659208779289795543733028307854060011893445184}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{6235763067481463703303197072852968997507774980887871613883060475893289445678145365515987946930552273368117333669804403132218153303355303855442719328}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{307421721130900228589719373934705836156991774018661550594199596392383243556275769428423217629549507038568109962009974162880622828138932659066859166903221404160}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{5091233710729124692807598192675197289555598293972987089450682995766513171963883788645935871887853391044050803623896388943274877797882115021344620543232}{269818500925236779930653620362719616431828188490599989546006204445287630699339328687659125894656939036685}, -\frac{52567320498344578115797908871589631859500613108375455982982191659102602143685025199811465194156867379891494317993827387944216159677146480384660}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{2389496970923777876301175592866182359398685398158451474005958930923247077940444247450306704054859526119246749957194841714748438373432808031584146241475399040}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{34572905616024395538457258468533256752563016817407427613240833698704830379286768052514182358939665107813360146077715825924315565501757384675560000}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{5671074928575348494343658731191545886040041382348418773431155118935585876027569273120472141406755624811526874973607837498773550004585943175598709565275797877760}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{11701983117470587325146787061658141630385685028044506614391809035837381521772942323743798097276097060927705578298578287634395927186894067365366390722293}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{2852614973426509493423483010436790242048397249204821078700768583366485314235621469482843679238789513299710196931208769755225144863526317761143129649638999712}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{42570323347378906135870223283181081865213041316742088897999716924865308822735188488081261703464801173427988708985214014685194767577572259156239646854960484933120}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{27649101588887347338673768336410051400382813491556908717874019563467839772412096192351616440173290489413349957534591250412143090045281840102808080045658}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, \frac{10240311887622542610382813921552329947433463558442127537984191062686056626903700056727884125599039172022481482636090196651644254377425239480136029381135391680}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}, -\frac{184211152394447298918388404653507343813772831617715917205111144787241152517894538158123375245848507096543449997939923221090395134726190405494624192766075007708160}{53963700185047355986130724072543923286365637698119997909201240889057526139867865737531825178931387807337}\right] \]

_{List(Fraction(Integer))}

_{Time: 0 sec}

No denominator is divisible by 11.

zdens := [zero? positiveRemainder(denom x, 11) for x in cfs]
assertEquals(removeDuplicates zdens, [false])

\[ \left[\texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}\right] \]

_{List(Boolean)}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

There are, however, some numerators that are not divisible by 11, so the relation does not reveal a divisibility by 11 for p(11n+6).

znums := [zero? positiveRemainder(numer x, 11) for x in cfs]
assertTrue(member?(false, znums))

\[ \left[\texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{false}, \texttt{true}, \texttt{true}, \texttt{false}, \texttt{true}, \texttt{false}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{false}, \texttt{true}, \texttt{false}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{false}, \texttt{true}, \texttt{false}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}, \texttt{true}\right] \]

_{List(Boolean)}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

-------------------------------------------------------------------
--endtest
-------------------------------------------------------------------

The relation in $M_i$'s with integer coefficients

-----------------------------------------------------------------
--test:integer
-----------------------------------------------------------------

Here $\mathbb{Z}$ is considered as a Euclidean domain wrt the absolute value as the Euclidean size function.

C ==> ZZ
--LZ ==> A1 ZZ
--XZ ==> X1 ZZ
nn := 22;
idxs := etaFunctionIndices nn;
mspecs := mSPECSInfM0(nn,idxs);
egens := [specM0A1(C)(spec) for spec in mspecs]
msyms := indexedSymbols("M", #egens)
xgens := [toX1(C, x, s) for x in egens for s in msyms]
 -- We only achieve a relation that shows divisibility by 11 with xgens.3.
xt := xgens.3
xab := samba(cons(xt, xgens))$QSAMBA(C,X1,QTOPRED)
assertEquals(# basis xab, 5)

_Void

_{Time: 0 sec}

_{PositiveInteger}

_{Time: 0 sec}

_{List(List(Integer))}

_{Time: 0 sec}

_{List(QEtaSpecification)}

_{Time: 0.02 (EV) = 0.02 sec}

\[ \left[{q}^{-5}+4\, {q}^{-4}+6\, {q}^{-3}+8\, {q}^{-2}+13\, {q}^{-1}+12+14\, q+24\, {q}^{2}+18\, {q}^{3}+20\, {q}^{4}+32\, {q}^{5}+O\left({q}^{6}\right), {q}^{-5}+{q}^{-4}+{q}^{-3}+2\, {q}^{-2}+2\, {q}^{-1}+3+4\, q+5\, {q}^{2}+6\, {q}^{3}+8\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), {q}^{-5}-7\, {q}^{-4}+17\, {q}^{-3}-14\, {q}^{-2}+2\, {q}^{-1}-21+36\, q+13\, {q}^{2}-26\, {q}^{3}-24\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), {q}^{-6}+2\, {q}^{-5}-{q}^{-4}-2\, {q}^{-3}-{q}^{-2}-6\, {q}^{-1}+4\, q-5\, {q}^{2}+4\, {q}^{3}+6\, {q}^{4}+O\left({q}^{5}\right), {q}^{-7}-4\, {q}^{-5}+2\, {q}^{-3}+8\, {q}^{-1}-5\, q-4\, {q}^{3}+O\left({q}^{4}\right), {q}^{-8}-2\, {q}^{-7}-3\, {q}^{-6}+6\, {q}^{-5}+2\, {q}^{-4}-{q}^{-2}-10\, {q}^{-1}-2\, q+10\, {q}^{2}+O\left({q}^{3}\right), {q}^{-9}-4\, {q}^{-8}+2\, {q}^{-7}+8\, {q}^{-6}-5\, {q}^{-5}-4\, {q}^{-4}-10\, {q}^{-3}+8\, {q}^{-2}+9\, {q}^{-1}+14\, q+O\left({q}^{2}\right)\right] \]

_{List(ModularFunctionQSeriesInfinity(Integer))}

_{Time: 0 sec}

\[ \left[M1, M2, M3, M4, M5, M6, M7\right] \]

_List(Symbol)

_{Time: 0 sec}

\[ \left[\left[{q}^{-5}+4\, {q}^{-4}+6\, {q}^{-3}+8\, {q}^{-2}+13\, {q}^{-1}+12+14\, q+24\, {q}^{2}+18\, {q}^{3}+20\, {q}^{4}+32\, {q}^{5}+O\left({q}^{6}\right), M1\right], \left[{q}^{-5}+{q}^{-4}+{q}^{-3}+2\, {q}^{-2}+2\, {q}^{-1}+3+4\, q+5\, {q}^{2}+6\, {q}^{3}+8\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), M2\right], \left[{q}^{-5}-7\, {q}^{-4}+17\, {q}^{-3}-14\, {q}^{-2}+2\, {q}^{-1}-21+36\, q+13\, {q}^{2}-26\, {q}^{3}-24\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), M3\right], \left[{q}^{-6}+2\, {q}^{-5}-{q}^{-4}-2\, {q}^{-3}-{q}^{-2}-6\, {q}^{-1}+4\, q-5\, {q}^{2}+4\, {q}^{3}+6\, {q}^{4}+O\left({q}^{5}\right), M4\right], \left[{q}^{-7}-4\, {q}^{-5}+2\, {q}^{-3}+8\, {q}^{-1}-5\, q-4\, {q}^{3}+O\left({q}^{4}\right), M5\right], \left[{q}^{-8}-2\, {q}^{-7}-3\, {q}^{-6}+6\, {q}^{-5}+2\, {q}^{-4}-{q}^{-2}-10\, {q}^{-1}-2\, q+10\, {q}^{2}+O\left({q}^{3}\right), M6\right], \left[{q}^{-9}-4\, {q}^{-8}+2\, {q}^{-7}+8\, {q}^{-6}-5\, {q}^{-5}-4\, {q}^{-4}-10\, {q}^{-3}+8\, {q}^{-2}+9\, {q}^{-1}+14\, q+O\left({q}^{2}\right), M7\right]\right] \]

_{List(QEtaExtendedAlgebra(Integer,ModularFunctionQSeriesInfinity(Integer),QEtaLazyAlgebra(Integer,Polynomial(Integer))))}

_{Time: 0 sec}

\[ \left[{q}^{-5}-7\, {q}^{-4}+17\, {q}^{-3}-14\, {q}^{-2}+2\, {q}^{-1}-21+36\, q+13\, {q}^{2}-26\, {q}^{3}-24\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), M3\right] \]

_{QEtaExtendedAlgebra(Integer,ModularFunctionQSeriesInfinity(Integer),QEtaLazyAlgebra(Integer,Polynomial(Integer)))}

_{Time: 0 sec}

-- numOfGaps:=[216, -1]
-- numOfGaps:=[216, -1]
-- numOfGaps:=[216, -1]
-- numOfGaps:=[216, -1]
-- numOfGaps:=[216, -1]
-- numOfGaps:=[162, -1]
-- numOfGaps:=[162, -1]
-- numOfGaps:=[109, -1]
-- numOfGaps:=[109, -1]
-- numOfGaps:=[56, -1]
-- numOfGaps:=[56, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]

\[ \left[mul=\left[{q}^{-5}-7\, {q}^{-4}+17\, {q}^{-3}-14\, {q}^{-2}+2\, {q}^{-1}-21+36\, q+13\, {q}^{2}-26\, {q}^{3}-24\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), M3\right], be=\texttt{table}\left(4=\left[\left[{q}^{-4}+31\, {q}^{-3}+2\, {q}^{-2}+33\, {q}^{-1}+3+62\, q+65\, {q}^{2}+4\, {q}^{3}+4\, {q}^{4}+66\, {q}^{5}+65\, {q}^{6}+O\left({q}^{7}\right), M3-4\, M2+3\, M1\right]\right], 3=\left[\left[88\, {q}^{-3}+88\, {q}^{-1}+176\, q+176\, {q}^{2}+176\, {q}^{5}+176\, {q}^{6}+88\, {q}^{7}+O\left({q}^{8}\right), 3\, M3-11\, M2+8\, M1\right], \left[{q}^{-8}-2\, {q}^{-7}-3\, {q}^{-6}+6\, {q}^{-5}+2\, {q}^{-4}-{q}^{-2}-10\, {q}^{-1}-2\, q+10\, {q}^{2}+O\left({q}^{3}\right), M6\right]\right], 2=\left[\left[{q}^{-7}-4\, {q}^{-5}+2\, {q}^{-3}+8\, {q}^{-1}-5\, q-4\, {q}^{3}+O\left({q}^{4}\right), M5\right]\right], 1=\left[\left[{q}^{-6}+2\, {q}^{-5}-{q}^{-4}-2\, {q}^{-3}-{q}^{-2}-6\, {q}^{-1}+4\, q-5\, {q}^{2}+4\, {q}^{3}+6\, {q}^{4}+O\left({q}^{5}\right), M4\right]\right]\right)\right] \]

_{QEtaAlgebraBasis(QEtaExtendedAlgebra(Integer,ModularFunctionQSeriesInfinity(Integer),QEtaLazyAlgebra(Integer,Polynomial(Integer))))}

_{Time: 0.01 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

rspec := eqSPEC(1,[-1]); nn := 22; m := 11; t := 6;
sspec: SPEC := cofactInfM0(nn, idxs, rspec, m, t)
assertEquals(sspec, eqSPEC(nn,[10,2,11,-22]))
xf := toX1(C, specM0A1(C)(sspec,rspec,m,t), F)

_{PositiveInteger}

_{Time: 0 sec}

-- == z:=[zinhom=[], zhom=[], zfree=[]]
-- >= z:=[zinhom=[[4, -1, 1], [5, -1, 2], [4, -1, 2]], zhom=[[3, -1, 0], [4, -1, 1], [1, -1, 1], [2, -1, 1], [3, -1, 1], [0, -1, 1], [2, 0, 1]], zfree=[]]

\[ \left[\left[1, 10\right], \left[2, 2\right], \left[11, 11\right], \left[22, -22\right]\right] \]

_{QEtaSpecification}

_{Time: 0.08 (EV) = 0.08 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

\[ \left[11\, {q}^{-14}+187\, {q}^{-13}+1111\, {q}^{-12}+3696\, {q}^{-11}+10164\, {q}^{-10}+22286\, {q}^{-9}+44396\, {q}^{-8}+79211\, {q}^{-7}+135608\, {q}^{-6}+214500\, {q}^{-5}+327415\, {q}^{-4}+O\left({q}^{-3}\right), F\right] \]

_{QEtaExtendedAlgebra(Integer,ModularFunctionQSeriesInfinity(Integer),QEtaLazyAlgebra(Integer,Polynomial(Integer)))}

_{Time: 0 sec}

Interestingly, we have to multiply by 4, because otherwise the reduction would not return 0.

xr := reduce(4*xf, xab)$QTOPRED(ZZ,X1)
assertTrue(zero? xr)

\[ \left[O\left({q}^{22}\right), \left(-19800\, M3-19081216\right)\, M5+\left(-51656\, M3+71554560\right)\, M4-44\, {M3}^{3}+\left(176\, M2-132\, M1-137819\right)\, {M3}^{2}+\left(556963\, M2-814880\, M1-16417280\right)\, M3-413227584\, M2+113124352\, M1+4\, F-524733440\right] \]

_{QEtaExtendedAlgebra(Integer,ModularFunctionQSeriesInfinity(Integer),QEtaLazyAlgebra(Integer,Polynomial(Integer)))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

pol := x1Pol(C,xr)
polExpected := (11*(_
    (-1800*M3-1734656)*M5_
    + (-4696*M3+6504960)*M4_
    - 4*M3^3_
    + (16*M2-12*M1-12529)*M3^2_
    + (50633*M2-74080*M1-1492480)*M3_
    - 37566144*M2_
    + 10284032*M1_
    - 47703040)_
  + 4*F)$PolZ
assertEquals(pol, polExpected)

\[ \left(-19800\, M3-19081216\right)\, M5+\left(-51656\, M3+71554560\right)\, M4-44\, {M3}^{3}+\left(176\, M2-132\, M1-137819\right)\, {M3}^{2}+\left(556963\, M2-814880\, M1-16417280\right)\, M3-413227584\, M2+113124352\, M1+4\, F-524733440 \]

_{Polynomial(Integer)}

_{Time: 0 sec}

\[ \left(-19800\, M3-19081216\right)\, M5+\left(-51656\, M3+71554560\right)\, M4-44\, {M3}^{3}+\left(176\, M2-132\, M1-137819\right)\, {M3}^{2}+\left(556963\, M2-814880\, M1-16417280\right)\, M3-413227584\, M2+113124352\, M1+4\, F-524733440 \]

_{Polynomial(Integer)}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

lcxr := [leadingCoefficient x for x in monomials(pol-4*F)];
divisibility11 := removeDuplicates [positiveRemainder(x, 11) for x in lcxr]
assertEquals(divisibility11, [0])

_{List(Integer)}

_{Time: 0 sec}

\[ \left[0\right] \]

_{List(Integer)}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

e := eval(pol, c+->c::LZZ, cons('F, msyms), cons(first xf, egens))_
     $ PolynomialEvaluation(ZZ, A1(ZZ))
assertTrue(zero? e)

\[ O\left({q}^{6}\right) \]

_{ModularFunctionQSeriesInfinity(Integer)}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

-------------------------------------------------------------------
--endtest
-------------------------------------------------------------------

Paule's proof via computation in $C[t,F]$

-----------------------------------------------------------------
--test:Paule
-----------------------------------------------------------------

rspec := eqSPEC(1,[-1]); nn := 11; m := 11; t := 6;
idxs := etaFunctionIndices nn
sspec := cofactInfM0(nn, idxs, rspec, m, t)
xf := toX1(C, specM0A1(C)(sspec,rspec,m,t), 'F)
xt := toX1(C, specM0A1(C) eqSPEC(11,[12,-12]), 'T)
xab := samba([xt, xf])$QSAMBA(C,X1,QTOPRED);
assertEquals(# basis xab, 4)
pols := sort((x,y)+->degree x < degree y, [x1Pol(C,x) for x in basis xab])
polsExpected := [F, F^2, F^3, F^4]$List(PolC)
assertEquals(pols, polsExpected)

_{PositiveInteger}

_{Time: 0 sec}

\[ \left[\left[1\right], \left[11\right]\right] \]

_{List(List(Integer))}

_{Time: 0 sec}

-- == z:=[zinhom=[[0]], zhom=[], zfree=[]]

\[ \left[\left[1, 12\right], \left[11, -11\right]\right] \]

_{QEtaSpecification}

_{Time: 0.03 (EV) = 0.03 sec}

\[ \left[11\, {q}^{-4}+165\, {q}^{-3}+748\, {q}^{-2}+1639\, {q}^{-1}+3553+4136\, q+6347\, {q}^{2}+3586\, {q}^{3}+7414\, {q}^{4}-4444\, {q}^{5}+583\, {q}^{6}+O\left({q}^{7}\right), F\right] \]

_{QEtaExtendedAlgebra(Integer,ModularFunctionQSeriesInfinity(Integer),QEtaLazyAlgebra(Integer,Polynomial(Integer)))}

_{Time: 0 sec}

\[ \left[{q}^{-5}-12\, {q}^{-4}+54\, {q}^{-3}-88\, {q}^{-2}-99\, {q}^{-1}+540-418\, q-648\, {q}^{2}+594\, {q}^{3}+836\, {q}^{4}+1056\, {q}^{5}+O\left({q}^{6}\right), T\right] \]

_{QEtaExtendedAlgebra(Integer,ModularFunctionQSeriesInfinity(Integer),QEtaLazyAlgebra(Integer,Polynomial(Integer)))}

_{Time: 0 sec}

-- numOfGaps:=[120, -1]
-- numOfGaps:=[120, -1]
-- numOfGaps:=[120, -1]
-- numOfGaps:=[120, -1]
-- numOfGaps:=[120, -1]
-- numOfGaps:=[120, -1]
-- numOfGaps:=[120, -1]
-- numOfGaps:=[120, -1]
-- numOfGaps:=[120, -1]
-- numOfGaps:=[120, -1]
-- numOfGaps:=[120, -1]
-- numOfGaps:=[120, -1]

_{QEtaAlgebraBasis(QEtaExtendedAlgebra(Integer,ModularFunctionQSeriesInfinity(Integer),QEtaLazyAlgebra(Integer,Polynomial(Integer))))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

\[ \left[F, {F}^{2}, {F}^{3}, {F}^{4}\right] \]

_{List(Polynomial(Integer))}

_{Time: 0 sec}

\[ \left[F, {F}^{2}, {F}^{3}, {F}^{4}\right] \]

_{List(Polynomial(Integer))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

xr := reduce(xf^5, xab)$QTOPRED(C,X1)
assertTrue(zero? xr)

\[ \left[O\left({q}^{22}\right), -161051\, {T}^{4}+\left(-1800843\, F-28100500582\right)\, {T}^{3}+\left(-5447783\, {F}^{2}+301427561028\, F-310989720965990\right)\, {T}^{2}+\left(-3674891\, {F}^{3}-199353759330\, {F}^{2}-533532824042570\, F+918994597271443220\right)\, T+{F}^{5}-73205\, {F}^{4}+2143588810\, {F}^{3}-31384283767210\, {F}^{2}+229748649317860805\, F-672749994932560009201\right] \]

_{QEtaExtendedAlgebra(Integer,ModularFunctionQSeriesInfinity(Integer),QEtaLazyAlgebra(Integer,Polynomial(Integer)))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

pol := x1Pol(C,xr)
c := coefficient(pol, F, 5)
polExpected := (-F^5_
  + 5*11^4*F^4_
  + 11^4*(251*T-2*5*11^4)*F^3_
  + 11^3*(4093*T^2+2*3*5*11^5*31*T+2*5*11^9)*F^2_
  + 11^4*(3*41*T^3-2^2*3*11^3*1289*T^2+2*5*11^8*17*T-5*11^12)*F_
  + 11^5*(T+11^4)*(T^3+11^2*1321*T^2-3*7*11^7*T+11^11))$PolZ
assertEquals(-pol, polExpected)

\[ -161051\, {T}^{4}+\left(-1800843\, F-28100500582\right)\, {T}^{3}+\left(-5447783\, {F}^{2}+301427561028\, F-310989720965990\right)\, {T}^{2}+\left(-3674891\, {F}^{3}-199353759330\, {F}^{2}-533532824042570\, F+918994597271443220\right)\, T+{F}^{5}-73205\, {F}^{4}+2143588810\, {F}^{3}-31384283767210\, {F}^{2}+229748649317860805\, F-672749994932560009201 \]

_{Polynomial(Integer)}

_{Time: 0 sec}

\[ 1 \]

_{Polynomial(Integer)}

_{Time: 0 sec}

\[ 161051\, {T}^{4}+\left(1800843\, F+28100500582\right)\, {T}^{3}+\left(5447783\, {F}^{2}-301427561028\, F+310989720965990\right)\, {T}^{2}+\left(3674891\, {F}^{3}+199353759330\, {F}^{2}+533532824042570\, F-918994597271443220\right)\, T-{F}^{5}+73205\, {F}^{4}-2143588810\, {F}^{3}+31384283767210\, {F}^{2}-229748649317860805\, F+672749994932560009201 \]

_{Polynomial(Integer)}

_{Time: 0.01 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

The above polynomial agrees with the relation that Peter Paule has on his Slide 32 of hia talk at "The 2016 International Number Theory Conference in honor of Krishna Alladi's 60th birthday" (Gainesville, March 17-21, 2016). Compare with formula (8) in \cite{Hemmecke::DancingSambaRamanujan:2018}.

-------------------------------------------------------------------
--endtest
-------------------------------------------------------------------

With $\mathbb{Z}_{(11)}$ and division by 11

-----------------------------------------------------------------
--test:Hemmecke2
-----------------------------------------------------------------

TODO: Introduce vectors in the second component so that we can track whether the just reduced element is divisible by 11.

C ==> IntegerLocalizedAtPrime 11
nn := 22::PP; -- level
idxs := etaFunctionIndices nn
m := 11::PP;
rspec: SPEC := eqSPEC(1,[-1])
orbs := [orb for tt in 0..m-1 | (orb := ORBIT(rspec,m,tt); one? # orb)]
t := first(first orbs) :: NN

_Void

_{Time: 0 sec}

_{PositiveInteger}

_{Time: 0 sec}

\[ \left[\left[1\right], \left[2\right], \left[11\right], \left[22\right]\right] \]

_{List(List(Integer))}

_{Time: 0 sec}

_{PositiveInteger}

_{Time: 0 sec}

\[ \left[\left[1, -1\right]\right] \]

_{QEtaSpecification}

_{Time: 0 sec}

\[ \left[\left[6\right]\right] \]

_{List(List(NonNegativeInteger))}

_{Time: 0 sec}

\[ 6 \]

_{NonNegativeInteger}

_{Time: 0 sec}

Let's compute the expansion (at $\infty$) of $$ F(\tau) = g_r(\tau) q^{\frac{e}{24}} \sum_{k=0}^{\infty}p(m k + t)q^k. $$

Create modular series for p(11n+6)

sspec := cofactInfM0(nn, idxs, rspec, m, t)
xf := toX1(C, specM0A1(C)(sspec,rspec,m,t), 'F)

-- == z:=[zinhom=[], zhom=[], zfree=[]]
-- >= z:=[zinhom=[[4, -1, 1], [5, -1, 2], [4, -1, 2]], zhom=[[3, -1, 0], [4, -1, 1], [1, -1, 1], [2, -1, 1], [3, -1, 1], [0, -1, 1], [2, 0, 1]], zfree=[]]

\[ \left[\left[1, 10\right], \left[2, 2\right], \left[11, 11\right], \left[22, -22\right]\right] \]

_{QEtaSpecification}

_{Time: 0.06 (EV) = 0.07 sec}

\[ \left[11\, {q}^{-14}+11\, 17\, {q}^{-13}+11\, 101\, {q}^{-12}+11\, 336\, {q}^{-11}+{11}^{2}\, 84\, {q}^{-10}+11\, 2026\, {q}^{-9}+11\, 4036\, {q}^{-8}+11\, 7201\, {q}^{-7}+11\, 12328\, {q}^{-6}+11\, 19500\, {q}^{-5}+11\, 29765\, {q}^{-4}+O\left({q}^{-3}\right), F\right] \]

_{QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11))))}

_{Time: 0 sec}

The generators of the eta-quotients of level 22 are given by these exponents for the eta-functions (with arguments being the divisors of 22). The variable mspecs corresponds to the generators of $R^\infty(22)$.

mspecs := mSPECSInfM0(nn, idxs);
egens := [specM0A1(C)(spec) for spec in mspecs]
msyms := indexedSymbols("M", #mspecs)
xgens := [toX1(C, x, sym) for x in egens for sym in msyms];
xgens := cons(toX1(C, egens.1, 'T), rest xgens);

_{List(QEtaSpecification)}

_{Time: 0.02 (EV) = 0.02 sec}

\[ \left[{q}^{-5}+4\, {q}^{-4}+6\, {q}^{-3}+8\, {q}^{-2}+13\, {q}^{-1}+12+14\, q+24\, {q}^{2}+18\, {q}^{3}+20\, {q}^{4}+32\, {q}^{5}+O\left({q}^{6}\right), {q}^{-5}+{q}^{-4}+{q}^{-3}+2\, {q}^{-2}+2\, {q}^{-1}+3+4\, q+5\, {q}^{2}+6\, {q}^{3}+8\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), {q}^{-5}-7\, {q}^{-4}+17\, {q}^{-3}-14\, {q}^{-2}+2\, {q}^{-1}-21+36\, q+13\, {q}^{2}-26\, {q}^{3}-24\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), {q}^{-6}+2\, {q}^{-5}-{q}^{-4}-2\, {q}^{-3}-{q}^{-2}-6\, {q}^{-1}+4\, q-5\, {q}^{2}+4\, {q}^{3}+6\, {q}^{4}+O\left({q}^{5}\right), {q}^{-7}-4\, {q}^{-5}+2\, {q}^{-3}+8\, {q}^{-1}-5\, q-4\, {q}^{3}+O\left({q}^{4}\right), {q}^{-8}-2\, {q}^{-7}-3\, {q}^{-6}+6\, {q}^{-5}+2\, {q}^{-4}-{q}^{-2}-10\, {q}^{-1}-2\, q+10\, {q}^{2}+O\left({q}^{3}\right), {q}^{-9}-4\, {q}^{-8}+2\, {q}^{-7}+8\, {q}^{-6}-5\, {q}^{-5}-4\, {q}^{-4}-10\, {q}^{-3}+8\, {q}^{-2}+9\, {q}^{-1}+14\, q+O\left({q}^{2}\right)\right] \]

_{List(ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)))}

_{Time: 0 sec}

\[ \left[M1, M2, M3, M4, M5, M6, M7\right] \]

_List(Symbol)

_{Time: 0 sec}

_{List(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0 sec}

_{List(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0 sec}

xt := xgens.1
xab := samba(xgens)$QSAMBA(C,X1,QTOPRED)

\[ \left[{q}^{-5}+4\, {q}^{-4}+6\, {q}^{-3}+8\, {q}^{-2}+13\, {q}^{-1}+12+14\, q+24\, {q}^{2}+18\, {q}^{3}+20\, {q}^{4}+32\, {q}^{5}+O\left({q}^{6}\right), T\right] \]

_{QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11))))}

_{Time: 0 sec}

-- numOfGaps:=[216, -1]
-- numOfGaps:=[216, -1]
-- numOfGaps:=[162, -1]
-- numOfGaps:=[162, -1]
-- numOfGaps:=[109, -1]
-- numOfGaps:=[109, -1]
-- numOfGaps:=[56, -1]
-- numOfGaps:=[56, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]

\[ \left[mul=\left[{q}^{-5}+4\, {q}^{-4}+6\, {q}^{-3}+8\, {q}^{-2}+13\, {q}^{-1}+12+14\, q+24\, {q}^{2}+18\, {q}^{3}+20\, {q}^{4}+32\, {q}^{5}+O\left({q}^{6}\right), T\right], be=\texttt{table}\left(4=\left[\left[{q}^{-4}+\frac{5}{3}\, {q}^{-3}+2\, {q}^{-2}+11\, \frac{1}{3}\, {q}^{-1}+3+\frac{10}{3}\, q+\frac{19}{3}\, {q}^{2}+4\, {q}^{3}+4\, {q}^{4}+11\, \frac{2}{3}\, {q}^{5}+\frac{19}{3}\, {q}^{6}+O\left({q}^{7}\right), \frac{1}{3}\, T-\frac{1}{3}\, M2\right]\right], 3=\left[\left[11\, {q}^{-3}+11\, {q}^{-1}+11\, 2\, q+11\, 2\, {q}^{2}+11\, 2\, {q}^{5}+11\, 2\, {q}^{6}+11\, {q}^{7}+O\left({q}^{8}\right), T+\frac{3}{8}\, M3+11\, \left(-\frac{1}{8}\right)\, M2\right], \left[{q}^{-8}-2\, {q}^{-7}-3\, {q}^{-6}+6\, {q}^{-5}+2\, {q}^{-4}-{q}^{-2}-10\, {q}^{-1}-2\, q+10\, {q}^{2}+O\left({q}^{3}\right), M6\right]\right], 2=\left[\left[{q}^{-7}-4\, {q}^{-5}+2\, {q}^{-3}+8\, {q}^{-1}-5\, q-4\, {q}^{3}+O\left({q}^{4}\right), M5\right]\right], 1=\left[\left[{q}^{-6}+2\, {q}^{-5}-{q}^{-4}-2\, {q}^{-3}-{q}^{-2}-6\, {q}^{-1}+4\, q-5\, {q}^{2}+4\, {q}^{3}+6\, {q}^{4}+O\left({q}^{5}\right), M4\right]\right]\right)\right] \]

_{QEtaAlgebraBasis(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0 sec}

Due to another critical element selection strategy, the resulting polynomial is a bit different.

be := basisElements xab
bas := sort(smallerGrade?, basis xab);

\[ \texttt{table}\left(4=\left[\left[{q}^{-4}+\frac{5}{3}\, {q}^{-3}+2\, {q}^{-2}+11\, \frac{1}{3}\, {q}^{-1}+3+\frac{10}{3}\, q+\frac{19}{3}\, {q}^{2}+4\, {q}^{3}+4\, {q}^{4}+11\, \frac{2}{3}\, {q}^{5}+\frac{19}{3}\, {q}^{6}+O\left({q}^{7}\right), \frac{1}{3}\, T-\frac{1}{3}\, M2\right]\right], 3=\left[\left[11\, {q}^{-3}+11\, {q}^{-1}+11\, 2\, q+11\, 2\, {q}^{2}+11\, 2\, {q}^{5}+11\, 2\, {q}^{6}+11\, {q}^{7}+O\left({q}^{8}\right), T+\frac{3}{8}\, M3+11\, \left(-\frac{1}{8}\right)\, M2\right], \left[{q}^{-8}-2\, {q}^{-7}-3\, {q}^{-6}+6\, {q}^{-5}+2\, {q}^{-4}-{q}^{-2}-10\, {q}^{-1}-2\, q+10\, {q}^{2}+O\left({q}^{3}\right), M6\right]\right], 2=\left[\left[{q}^{-7}-4\, {q}^{-5}+2\, {q}^{-3}+8\, {q}^{-1}-5\, q-4\, {q}^{3}+O\left({q}^{4}\right), M5\right]\right], 1=\left[\left[{q}^{-6}+2\, {q}^{-5}-{q}^{-4}-2\, {q}^{-3}-{q}^{-2}-6\, {q}^{-1}+4\, q-5\, {q}^{2}+4\, {q}^{3}+6\, {q}^{4}+O\left({q}^{5}\right), M4\right]\right]\right) \]

_{XHashTable(Integer,List(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11))))))}

_{Time: 0 sec}

_{List(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0 sec}

We show that the basis element of degree 3 has coefficients that are all divisible by 11. So we can divide it by 11. The method is to multiply by t and track the cofactors of the basis elements used in this reduction. That this reduction should give 0 is clear, since xab is an algebra basis.

x3 := xt*be.3.1 - 11*(be.3.2 + 6*be.2.1 +10*be.1.1 + 10*xt - 11*be.4.1 - ((1/3)::C)*be.3.1 -11::C*1$X)
assertTrue(zero? x3)

\[ \left[O\left({q}^{13}\right), {T}^{2}+\left(\frac{3}{8}\, M3+11\, \left(-\frac{1}{8}\right)\, M2+11\, \left(-6\right)\right)\, T+11\, \left(-1\right)\, M6+11\, \left(-6\right)\, M5+11\, \left(-10\right)\, M4+11\, \frac{1}{8}\, M3+{11}^{2}\, \left(-\frac{3}{8}\right)\, M2+{11}^{2}\right] \]

_{QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11))))}

_{Time: 0.02 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

It's still not a representation in terms of the Mi.

p := x1Pol(C,x3)
l := [[e, x] for x in monomials(p)| zero?(e:=exponent leadingCoefficient x)]
assertTrue(not empty? l)

\[ {T}^{2}+\left(\frac{3}{8}\, M3+11\, \left(-\frac{1}{8}\right)\, M2+11\, \left(-6\right)\right)\, T+11\, \left(-1\right)\, M6+11\, \left(-6\right)\, M5+11\, \left(-10\right)\, M4+11\, \frac{1}{8}\, M3+{11}^{2}\, \left(-\frac{3}{8}\right)\, M2+{11}^{2} \]

_{Polynomial(IntegerLocalizedAtPrime(11))}

_{Time: 0 sec}

\[ \left[\left[0, {T}^{2}\right], \left[0, \frac{3}{8}\, M3\, T\right]\right] \]

_{List(List(Polynomial(IntegerLocalizedAtPrime(11))))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

-------------------------------------------------------------------
--endtest
-------------------------------------------------------------------

With $\mathbb{Z}_{(11)}$ and different strategy

-----------------------------------------------------------------
--test:Hemmecke3
-----------------------------------------------------------------

This test shows divisibility by 11, but uses a specifically tailored computation of an algebra basis and additional mamual reduction.

C ==> IntegerLocalizedAtPrime 11
nn := 22::PP; -- level
idxs := etaFunctionIndices nn
m := 11::PP;
rspec: SPEC := eqSPEC(1,[-1])
orbs := [orb for tt in 0..m-1 | (orb := ORBIT(rspec,m,tt); one? # orb)]
t := first(first orbs) :: NN

_Void

_{Time: 0 sec}

_{PositiveInteger}

_{Time: 0 sec}

\[ \left[\left[1\right], \left[2\right], \left[11\right], \left[22\right]\right] \]

_{List(List(Integer))}

_{Time: 0 sec}

_{PositiveInteger}

_{Time: 0 sec}

\[ \left[\left[1, -1\right]\right] \]

_{QEtaSpecification}

_{Time: 0 sec}

\[ \left[\left[6\right]\right] \]

_{List(List(NonNegativeInteger))}

_{Time: 0 sec}

\[ 6 \]

_{NonNegativeInteger}

_{Time: 0 sec}

Let's compute the expansion (at $\infty$) of $$ F(\tau) = g_r(\tau) q^{\frac{e}{24}} \sum_{k=0}^{\infty}p(m k + t)q^k. $$

Create modular series for p(11n+6)

sspec := cofactInfM0(nn, idxs, rspec, m, t)
xf := toX1(C, specM0A1(C)(sspec,rspec,m,t), 'F)

-- == z:=[zinhom=[], zhom=[], zfree=[]]
-- >= z:=[zinhom=[[4, -1, 1], [5, -1, 2], [4, -1, 2]], zhom=[[3, -1, 0], [4, -1, 1], [1, -1, 1], [2, -1, 1], [3, -1, 1], [0, -1, 1], [2, 0, 1]], zfree=[]]

\[ \left[\left[1, 10\right], \left[2, 2\right], \left[11, 11\right], \left[22, -22\right]\right] \]

_{QEtaSpecification}

_{Time: 0.08 (EV) = 0.08 sec}

\[ \left[11\, {q}^{-14}+11\, 17\, {q}^{-13}+11\, 101\, {q}^{-12}+11\, 336\, {q}^{-11}+{11}^{2}\, 84\, {q}^{-10}+11\, 2026\, {q}^{-9}+11\, 4036\, {q}^{-8}+11\, 7201\, {q}^{-7}+11\, 12328\, {q}^{-6}+11\, 19500\, {q}^{-5}+11\, 29765\, {q}^{-4}+O\left({q}^{-3}\right), F\right] \]

_{QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11))))}

_{Time: 0 sec}

The generators of the eta-quotients of level 22 are given by these exponents for the eta-functions (with arguments being the divisors of 22). The variable mspecs corresponds to the generators of $R^\infty(22)$.

mspecs := mSPECSInfM0(nn, idxs);
egens := [specM0A1(C)(spec) for spec in mspecs]
msyms := indexedSymbols("M", #mspecs)
xgens := [toX1(C, x, sym) for x in egens for sym in msyms];
xgens := cons(toX1(C, egens.1, 'T), rest xgens);

_{List(QEtaSpecification)}

_{Time: 0.02 (EV) = 0.02 sec}

\[ \left[{q}^{-5}+4\, {q}^{-4}+6\, {q}^{-3}+8\, {q}^{-2}+13\, {q}^{-1}+12+14\, q+24\, {q}^{2}+18\, {q}^{3}+20\, {q}^{4}+32\, {q}^{5}+O\left({q}^{6}\right), {q}^{-5}+{q}^{-4}+{q}^{-3}+2\, {q}^{-2}+2\, {q}^{-1}+3+4\, q+5\, {q}^{2}+6\, {q}^{3}+8\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), {q}^{-5}-7\, {q}^{-4}+17\, {q}^{-3}-14\, {q}^{-2}+2\, {q}^{-1}-21+36\, q+13\, {q}^{2}-26\, {q}^{3}-24\, {q}^{4}+10\, {q}^{5}+O\left({q}^{6}\right), {q}^{-6}+2\, {q}^{-5}-{q}^{-4}-2\, {q}^{-3}-{q}^{-2}-6\, {q}^{-1}+4\, q-5\, {q}^{2}+4\, {q}^{3}+6\, {q}^{4}+O\left({q}^{5}\right), {q}^{-7}-4\, {q}^{-5}+2\, {q}^{-3}+8\, {q}^{-1}-5\, q-4\, {q}^{3}+O\left({q}^{4}\right), {q}^{-8}-2\, {q}^{-7}-3\, {q}^{-6}+6\, {q}^{-5}+2\, {q}^{-4}-{q}^{-2}-10\, {q}^{-1}-2\, q+10\, {q}^{2}+O\left({q}^{3}\right), {q}^{-9}-4\, {q}^{-8}+2\, {q}^{-7}+8\, {q}^{-6}-5\, {q}^{-5}-4\, {q}^{-4}-10\, {q}^{-3}+8\, {q}^{-2}+9\, {q}^{-1}+14\, q+O\left({q}^{2}\right)\right] \]

_{List(ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)))}

_{Time: 0 sec}

\[ \left[M1, M2, M3, M4, M5, M6, M7\right] \]

_List(Symbol)

_{Time: 0 sec}

_{List(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0 sec}

_{List(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0 sec}

Because we observed that using all generators of grade 5 leads to a series (of grade 3) with leading coefficient 11 in the basis, we compute a basis with just one of those generators. Of course, we cannot expect that this leads to a basis by which we can reduce xf to zero. However, we hope that we arrive at a zero reduction if we later use the other grade 5 elements.

#xgens
xgens0 := [xgens.i for i in [1,4,5,6,7]]
xab0 := samba(xgens0)$QSAMBA(C,X1,QTOPRED)

\[ 7 \]

_{PositiveInteger}

_{Time: 0 sec}

\[ \left[\left[{q}^{-5}+4\, {q}^{-4}+6\, {q}^{-3}+8\, {q}^{-2}+13\, {q}^{-1}+12+14\, q+24\, {q}^{2}+18\, {q}^{3}+20\, {q}^{4}+32\, {q}^{5}+O\left({q}^{6}\right), T\right], \left[{q}^{-6}+2\, {q}^{-5}-{q}^{-4}-2\, {q}^{-3}-{q}^{-2}-6\, {q}^{-1}+4\, q-5\, {q}^{2}+4\, {q}^{3}+6\, {q}^{4}+O\left({q}^{5}\right), M4\right], \left[{q}^{-7}-4\, {q}^{-5}+2\, {q}^{-3}+8\, {q}^{-1}-5\, q-4\, {q}^{3}+O\left({q}^{4}\right), M5\right], \left[{q}^{-8}-2\, {q}^{-7}-3\, {q}^{-6}+6\, {q}^{-5}+2\, {q}^{-4}-{q}^{-2}-10\, {q}^{-1}-2\, q+10\, {q}^{2}+O\left({q}^{3}\right), M6\right], \left[{q}^{-9}-4\, {q}^{-8}+2\, {q}^{-7}+8\, {q}^{-6}-5\, {q}^{-5}-4\, {q}^{-4}-10\, {q}^{-3}+8\, {q}^{-2}+9\, {q}^{-1}+14\, q+O\left({q}^{2}\right), M7\right]\right] \]

_{List(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0 sec}

-- numOfGaps:=[216, -1]
-- numOfGaps:=[163, -1]
-- numOfGaps:=[110, -1]
-- numOfGaps:=[57, -1]
-- numOfGaps:=[4, -1]
-- numOfGaps:=[4, -1]
-- numOfGaps:=[4, -1]
-- numOfGaps:=[4, -1]
-- numOfGaps:=[4, -1]
-- numOfGaps:=[4, -1]
-- numOfGaps:=[4, -1]
-- numOfGaps:=[4, -1]
-- numOfGaps:=[4, -1]
-- numOfGaps:=[4, -1]
-- numOfGaps:=[4, -1]

\[ \left[mul=\left[{q}^{-5}+4\, {q}^{-4}+6\, {q}^{-3}+8\, {q}^{-2}+13\, {q}^{-1}+12+14\, q+24\, {q}^{2}+18\, {q}^{3}+20\, {q}^{4}+32\, {q}^{5}+O\left({q}^{6}\right), T\right], be=\texttt{table}\left(1=\left[\left[{q}^{-6}+2\, {q}^{-5}-{q}^{-4}-2\, {q}^{-3}-{q}^{-2}-6\, {q}^{-1}+4\, q-5\, {q}^{2}+4\, {q}^{3}+6\, {q}^{4}+O\left({q}^{5}\right), M4\right]\right], 3=\left[\left[{q}^{-8}-2\, {q}^{-7}-3\, {q}^{-6}+6\, {q}^{-5}+2\, {q}^{-4}-{q}^{-2}-10\, {q}^{-1}-2\, q+10\, {q}^{2}+O\left({q}^{3}\right), M6\right]\right], 4=\left[\left[{q}^{-9}-4\, {q}^{-8}+2\, {q}^{-7}+8\, {q}^{-6}-5\, {q}^{-5}-4\, {q}^{-4}-10\, {q}^{-3}+8\, {q}^{-2}+9\, {q}^{-1}+14\, q+O\left({q}^{2}\right), M7\right]\right], 2=\left[\left[{q}^{-7}-4\, {q}^{-5}+2\, {q}^{-3}+8\, {q}^{-1}-5\, q-4\, {q}^{3}+O\left({q}^{4}\right), M5\right]\right]\right)\right] \]

_{QEtaAlgebraBasis(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0 sec}

bas := sort(smallerGrade?, basis xab0);
[qetaGrade x for x in bas]
xr0 := reduce(xf,xab0)$QTOPRED(C,X1)

_{List(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0 sec}

\[ \left[6, 7, 8, 9\right] \]

_{List(Integer)}

_{Time: 0 sec}

\[ \left[{11}^{3}\, {q}^{-4}+{11}^{3}\, \frac{61}{53}\, {q}^{-3}+{11}^{3}\, 2\, {q}^{-2}+{11}^{3}\, \frac{167}{53}\, {q}^{-1}+{11}^{3}\, \frac{21}{53}+{11}^{3}\, \frac{122}{53}\, q+{11}^{3}\, \frac{281}{53}\, {q}^{2}+{11}^{3}\, 4\, {q}^{3}+{11}^{3}\, 4\, {q}^{4}+{11}^{3}\, \frac{334}{53}\, {q}^{5}+{11}^{3}\, \frac{281}{53}\, {q}^{6}+O\left({q}^{7}\right), {11}^{2}\, \frac{18}{53}\, {T}^{2}+\left(11\, \frac{1}{53}\, M7+11\, \frac{17}{53}\, M6+11\, \frac{75}{53}\, M5+{11}^{3}\, \frac{1}{53}\, M4+{11}^{2}\, \left(-\frac{1041}{53}\right)\right)\, T+{11}^{2}\, \left(-\frac{56}{53}\right)\, M7+{11}^{2}\, \left(-\frac{463}{53}\right)\, M6+{11}^{2}\, \left(-\frac{1399}{53}\right)\, M5+{11}^{2}\, \left(-\frac{2109}{53}\right)\, M4-\frac{1}{53}\, F\right] \]

_{QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11))))}

_{Time: 0 sec}

As was to be expected that reduction does not lead to zero.

xab := samba(xgens)$QSAMBA(C,X1,QTOPRED);
xr := reduce(xr0,xab)$QTOPRED(C,X1)

-- numOfGaps:=[216, -1]
-- numOfGaps:=[216, -1]
-- numOfGaps:=[162, -1]
-- numOfGaps:=[162, -1]
-- numOfGaps:=[109, -1]
-- numOfGaps:=[109, -1]
-- numOfGaps:=[56, -1]
-- numOfGaps:=[56, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]
-- numOfGaps:=[2, -1]

_{QEtaAlgebraBasis(QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11)))))}

_{Time: 0.01 sec}

\[ \left[O\left({q}^{22}\right), {11}^{2}\, \left(-\frac{3}{23}\right)\, {T}^{2}+\left(11\, \left(-\frac{1}{138}\right)\, M7+11\, \left(-\frac{17}{138}\right)\, M6+11\, \left(-\frac{25}{46}\right)\, M5+{11}^{3}\, \left(-\frac{1}{138}\right)\, M4+{11}^{2}\, \frac{604}{69}\right)\, T+{11}^{2}\, \frac{28}{69}\, M7+{11}^{2}\, \frac{463}{138}\, M6+{11}^{2}\, \frac{1399}{138}\, M5+{11}^{2}\, \frac{703}{46}\, M4+{11}^{2}\, \left(-\frac{41}{552}\right)\, M3+{11}^{3}\, \left(-\frac{19}{184}\right)\, M2+\frac{1}{138}\, F+{11}^{3}\, \left(-1\right)\right] \]

_{QEtaExtendedAlgebra(IntegerLocalizedAtPrime(11),ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11)),QEtaLazyAlgebra(IntegerLocalizedAtPrime(11),Polynomial(IntegerLocalizedAtPrime(11))))}

_{Time: 0 sec}

And, voila, we have a representation that reveals divisibility by 11.

pol := x1Pol(C, xr)
fpol := F - ((1/coefficient(pol,'F,1))::C)*pol
assertTrue(not member?('F, variables(fpol)))
c11s := [exponent leadingCoefficient m for m in monomials fpol]
assertEquals(removeDuplicates [x>0 for x in c11s], [true])

\[ {11}^{2}\, \left(-\frac{3}{23}\right)\, {T}^{2}+\left(11\, \left(-\frac{1}{138}\right)\, M7+11\, \left(-\frac{17}{138}\right)\, M6+11\, \left(-\frac{25}{46}\right)\, M5+{11}^{3}\, \left(-\frac{1}{138}\right)\, M4+{11}^{2}\, \frac{604}{69}\right)\, T+{11}^{2}\, \frac{28}{69}\, M7+{11}^{2}\, \frac{463}{138}\, M6+{11}^{2}\, \frac{1399}{138}\, M5+{11}^{2}\, \frac{703}{46}\, M4+{11}^{2}\, \left(-\frac{41}{552}\right)\, M3+{11}^{3}\, \left(-\frac{19}{184}\right)\, M2+\frac{1}{138}\, F+{11}^{3}\, \left(-1\right) \]

_{Polynomial(IntegerLocalizedAtPrime(11))}

_{Time: 0 sec}

\[ {11}^{2}\, 18\, {T}^{2}+\left(11\, M7+11\, 17\, M6+11\, 75\, M5+{11}^{3}\, M4+{11}^{2}\, \left(-1208\right)\right)\, T+{11}^{2}\, \left(-56\right)\, M7+{11}^{2}\, \left(-463\right)\, M6+{11}^{2}\, \left(-1399\right)\, M5+{11}^{2}\, \left(-2109\right)\, M4+{11}^{2}\, \frac{41}{4}\, M3+{11}^{3}\, \frac{57}{4}\, M2+{11}^{3}\, 138 \]

_{Polynomial(IntegerLocalizedAtPrime(11))}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

\[ \left[2, 1, 1, 1, 3, 2, 2, 2, 2, 2, 2, 3, 3\right] \]

_{List(NonNegativeInteger)}

_{Time: 0 sec}

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

syms := cons('T,rest msyms)

\[ \left[T, M2, M3, M4, M5, M6, M7\right] \]

_List(Symbol)

_{Time: 0 sec}

rf := eval(fpol, c+->c*1$A1(C),syms,egens)$PolynomialEvaluation(C,A1 C)

\[ 11\, {q}^{-14}+11\, 17\, {q}^{-13}+11\, 101\, {q}^{-12}+11\, 336\, {q}^{-11}+{11}^{2}\, 84\, {q}^{-10}+11\, 2026\, {q}^{-9}+11\, 4036\, {q}^{-8}+11\, 7201\, {q}^{-7}+11\, 12328\, {q}^{-6}+11\, 19500\, {q}^{-5}+11\, 29765\, {q}^{-4}+O\left({q}^{-3}\right) \]

_{ModularFunctionQSeriesInfinity(IntegerLocalizedAtPrime(11))}

_{Time: 0 sec}

assertTrue(zero?(x1A1(C,xf) - rf))

\[ \texttt{true} \]

_Boolean

_{Time: 0 sec}

-------------------------------------------------------------------
--endtest
-------------------------------------------------------------------