QModularGammaCategory¶

QModularCategory provides functions to check modularity conditions.

candidateLevelsCoEtaQuotient: (QEtaSpecification, PositiveInteger, NonNegativeInteger) -> List PositiveInteger: candidateLevelsCoEtaQuotient(rspec,m,t) returns an ascendingly sorted list of the nn up to 24*m*level(rspec) such that conditionCoEtaQuotient?(nn,rspec,m,t) is true.

conditionCoEtaQuotient?: (PositiveInteger, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> Boolean: conditionCoEtaQuotient?(nn,rspec,m,t) returns true iff a cofactor eta-quotient can be found such that the product of the cofactor with the orbit product corresponding to the dissection (m,t) gives a modular function in level nn.

minimalLevelCoEtaQuotient: (QEtaSpecification, PositiveInteger, NonNegativeInteger) -> PositiveInteger: minimalLevelCoEtaQuotient(rspec, m, t) returns the smallest nn of candidateLevalsCoEtaQuotient(rspec,m,t).

minimalLevelCoEtaQuotient: (QPochhammerSpecification, PositiveInteger, NonNegativeInteger) -> PositiveInteger: minimalLevelCoEtaQuotient(rspec, m, t) returns the smallest nn of candidateLevalsCoEtaQuotient(rspec,m,t).

minimalLevelCoEtaQuotient: QGeneratingFunctionVariable -> PositiveInteger: minimalLevelCoEtaQuotient(dissect) returns minimalLevelCoEtaQuotient(rspec,m,t) where rspec=definingSpecification(dissect), m=multiplier(dissect), and t=offset(dissect).

modularOrbit: (QEtaSpecification, PositiveInteger, NonNegativeInteger) -> List NonNegativeInteger: modularOrbit(rspec, m, t) (for $Gamma_0(N) where N is level(rspec)) computes the elements of $modularOrbit{r,m,t)$ as defined in qeta.tex, cite[Definition~42]{Radu_RamanujanKolberg_2015} and cite[Lemma~4.35]{Radu_PhD_2010} were r is pureExponents(rspec) and rspec is expected to specify a pure eta-quotient. For $Gamma_1(N)$, modularOrbit(rspec, m, t) returns [t].

modularOrbit: QGeneratingFunctionVariable -> List NonNegativeInteger: modularOrbit(dissect) returns modularOrbit(rspec,m,t) where rspec=definingSpecification(dissect), m=multiplier(dissect), and t=offset(dissect).