SymbolicModularEtaQuotientGamma QMODΒΆ
qetasymbmod.spad line 132 [edit on github]
- QMOD: QEtaModularCategory 
SymbolicModularEtaQuotientGamma(QMOD) holds data to compute an eta-quotient expansions of $F_{s,r,m,t}(gamma tau)$ for a particular matrix $gammainSL2Z$.. See eqref{eq:F_s-r-m-t(gamma*tau)}.
- basefactor: % -> List SymbolicEtaQuotientGamma
- basefactor(x)returns the part of $- F_{- s,- r,- m,- t}(gamma tau)$ that is connected to the generating series of a(m*n+k) for- kin modularOrbit(rspec,- m,- t) (from 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 $- F_{- s,- r,- m,- t}$ a modular function for $Gamma_0(- N)$ or $Gamma_1(- N)$ (depending on the parameter QMOD.
- etaQuotient: (QEtaSpecification, Matrix Integer) -> %
- generalizedEtaQuotient(rspec, gamma) represents the expansion of $ - g_r(gamma tau)$ where- ris given through rspec.
- etaQuotient: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger, Matrix Integer) -> %
- etaQuotient(sspec, rspec, m, t, gamma)represents the expansion of $- F_{- s,- r,- m,- t}(gamma tau)$ where- sand- rare given through- sspecand- rspec.
- etaQuotient: QEtaSpecification -> %
- etaQuotient(rspec)represents the expansion of $- g_r(tau)$ where- ris given through- rspec. It is the same as generalizedEtaQuotient(- rspec,matrix[[1,0],[0,1]]).
- hash: % -> SingleInteger
- from Hashable 
- hashUpdate!: (HashState, %) -> HashState
- from Hashable 
- latex: % -> String
- from SetCategory 
- minimalRootOfUnity: % -> PositiveInteger
- minimalRootOfUnity(x)returns the- lcmof minimalRootOfUnity(cofactor(- x)) and- lcm([minimalRootOfUnity- bfor- bin basefactor(- x)]).
- one?: % -> Boolean
- one?(x)returns- trueif the generalized eta-quotient corresponding to- xrepresents 1. This is the case if one?(first(basefactor(- x))) and one?(cofactor(- x)). If this function returns- false, then it is not guaranteed that- xdoes not represent the constant series 1; it might still be the case that basefactor(- x) represents the inverse of cofactor(- x).
- order: % -> Integer
- order(x)returns the order of the- z-expansion in terms of the z=q^(1/w) where- wis the width of the cusp of the group corresponding to QMOD that corresponds to transformationMatrix(- x). If the input condition multiplier(first(basefactor(- x)))- =1is not met, the function aborts with an error.
- orderMin: % -> Fraction Integer
- orderMin(x)returns the expected order of the- z-expansion in terms of z=q^(1/w) where- wis the width of the cusp of the group corresponding to QMOD that corresponds to transformationMatrix(- x). Note that this is a lower bound of the order. If u=ceiling(orderMin(- x)) then even this integer is a lower bound of the order of the expansion of- zin terms of x=q^(1/w), since the coefficient corresponding to x^u 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(first(basefactor(- x)))- =1orderMin(- x) is equal to the- trueorder of the expansion.
- qExponent: % -> Fraction Integer
- qExponent(x)returns the order of the- q-expansion in terms of the original- q. If the input condition multiplier(first(basefactor(- x)))- =1is not met, the function aborts with an error.
- 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. Only for cases where multiplier(first(basefactor(- x)))- =1qExponentMin(- x) is equal to the- trueorder (in- q) of the expansion.