SymbolicModularEtaQuotientGamma QMODΒΆ

qetasymbmod.spad line 130 [edit on github]

SymbolicModularEtaQuotientGamma(QMOD) holds data to compute an eta-quotient expansions of $F_{bar{s}, bar{r}, m, t}(gammatau)$. See eqref{eq:F_sbar-rbar-m-t(gamma*tau)}.

=: (%, %) -> Boolean

from BasicType

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from BasicType

basefactor: % -> List SymbolicEtaQuotientGamma

basefactor(x) returns the part of $F_{bar{s},bar{r},m,t}$ that is connected to the generating series of a(m*n+k) for k in modularOrbit(rspec,m,t)$QMOD.

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: SymbolicEtaQuotientGamma -> %

coerce(y) turns an element of SymbolicEtaQuotientGamma into this domain.

cofactor: % -> SymbolicEtaQuotientGamma

cofactor(x) returns the cofactor part to make $F_{bar{s},bar{r},m,t}$ a modular function for $Gamma_1(N)$.

etaQuotient: (QEtaSpecification, Matrix Integer) -> %

generalizedEtaQuotient(rspec, gamma) represents the expansion of $g_{bar{r}}(gammatau)$ where bar{r} is given through rspec.

etaQuotient: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger, Matrix Integer) -> %

etaQuotient(sspec, rspec, m, t, gamma) represents the expansion of $F_{bar{s},bar{r},m,t}(gammatau)$ where bar{s} and bar{r} are given through sspec and rspec.

etaQuotient: QEtaSpecification -> %

etaQuotient(rspec) represents the expansion of $g_{bar{r}}(tau)$ where bar{r} is given through rspec. It is the same as generalizedEtaQuotient(rspec,matrix[[1,0],[0,1]]).

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

latex: % -> String

from SetCategory

minimalRootOfUnity: % -> PositiveInteger

minimalRootOfUnity(x) returns the lcm of minimalRootOfUnity(cofactor(x)) and minimalRootOfUnity(basefactor(x)).

one?: % -> Boolean

one?(x) returns true if the generalized eta-quotient corresponding to x represents 1. This is the case if one?(basefactor(x)) and one?(cofactor(x)).

orderMin: % -> Fraction Integer

orderMin0(x) returns the expected order of the x-expansion in terms of x=q^(1/w) where w is the widths of the cusp of G corresponding to transformationMatrix(x). Note that this is an lower bound of the pole order. The coefficient corresponding to this grade may be zero. Further note that due to estimation it does not necessarily return an integer, but a rational number. Only for cases where multiplier(basefactor(x))=1 and one?(cofactor(x), orderMin(x) is equal to the true order of the expansion.

qExponentMin: % -> Fraction Integer

qExponentMin(x) returns the order of the q-expansion in terms of the original q. Note that this exponent is only a lower bound for the q-expansion. The coefficient corresponding to this q-power may be zero.

BasicType

CoercibleTo OutputForm

SetCategory