ModularEtaQuotientQSeriesTools(C, xiord, CX, xi, MODG, trfs)¶

qetamodeqqseriestool.spad line 133 [edit on github]

ModularEtaQuotientQSeriesTools computes an eta quotient expansion of $g_{r,m,lambda}(gamma tau)$, $p_{r,m,t}(gammatau)$, $F_{s,r,m,t}(gammatau)$. See eqref{eq:g_r-m-lambda(gamma*tau)} and eqref{eq:F_s-r-m-t(gamma*tau)}.

modularEtaQuotient: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> ModularFunctionQSeries(CX, trfs): modularEtaQuotient(sspec,rspec,m,t) returns the series expansion of etaQuotient(sspec,rspec,m,t)$SymbolicModularEtaQuotientGamma(MODG) at the respective cusps.

modularEtaQuotient: QEtaSpecification -> ModularFunctionQSeries(CX, trfs): modularEtaQuotient(rspec) checks whether rspec specifices modular eta-quotient wrt. the group represented by MODG and returns the Laurent expansion at the respective cusps. Due to Theorem 1 (Gordon, Hughes, Newman) in https://doi.org/10.1515/crll.1996.472.1 we only need to require that the weight of the modular eta-quotient is an integer and the other modular conditions are fulfilled, i.e. we only check that weakModular?(rspec) is true instead of modular?(rspec).

modularFunction: QEtaSpecificationExpression C -> ModularFunctionQSeries(CX, trfs): modularFunction(x) effectively replaces each specification in the term of x by the respective expansion at all the cusps of the group repesented by MODG.

modularFunction: QEtaSpecificationExpressionMonomial -> ModularFunctionQSeries(CX, trfs): modularFunction(x) returns the expansion of x at trfs if specificationMonomial(x)$SymbolicModularEtaQuotientGamma(MODG) does not lead to failure.