SymbolicEtaQuotientQSeriesTools(C, xiord, CX, xi, QMOD)ΒΆ
qetasymbeqqseriestool.spad line 128 [edit on github]
- xiord: PositiveInteger 
- CX: Algebra C 
- xi: CX 
- QMOD: QEtaModularCategory 
SymbolicEtaQuotientQSeriesTools 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)}.
- etaQuotientTraceExpansions: (QEtaSpecification, PositiveInteger, List Cusp) -> XHashTable(Matrix Integer, QEtaLaurentSeries CX)
- etaQuotientTraceExpansions(spec,mm,spitzen)returns etaQuotientTraceExpansions(- spec,- mm,gammas) for nn:=level(- spec) and gammas:=[cuspToMatrix(- nn,- c)$QMOD for- cin- spitzen].
- etaQuotientTraceExpansions: (QEtaSpecification, PositiveInteger, List Matrix Integer) -> XHashTable(Matrix Integer, QEtaLaurentSeries CX)
- etaQuotientTraceExpansions(spec,mm,gammas)returns expansions of the (modular- wrt. QMOD corresponding to- nn) eta-quotient given by- specat each gamma of- gammaswhere nn=level(- spec). See definition of the trace operator in cite{- Kohnen_WeierstrassPointsAtInfinity_2004}. Note that if level(- spec)- =mmthen etaQuotientTraceExpansions(- spec,- mm,- gammas) is equal to laurentExpansions(- y,- gammas) for y=etaQuotient(- spec,- gammas)$SymbolicModularEtaQuotient(QMOD).
- laurentExpansion: SymbolicModularEtaQuotientGamma QMOD -> QEtaLaurentSeries CX
- laurentExpansion(y)represents the- q-expansion of $- F_{- s,- r,- m,- t}(gammatau)$ given by- yin the canonical variables given by the width of the cusp- wrt. QMOD.
- laurentExpansionInfinity: SymbolicModularEtaQuotientGamma QMOD -> QEtaLaurentSeries CX
- laurentExpansionInfinity(y)represents the- q-expansion of $- F_{- s,- r,- m,- t}(tau)$ given by- y- wrt. QMOD. See eqref{eq:F_s-- r-- m-- t(tau)}.
- laurentExpansions: (SymbolicModularEtaQuotient QMOD, List Cusp) -> XHashTable(Matrix Integer, QEtaLaurentSeries CX)
- laurentExpansions(y, spitzen)returns laurentExpansions(- y,gammas) for nn:=level(- y) and gammas:=[cuspToMatrix(- nn,- c)$QMOD for- cin spitzen].
- laurentExpansions: (SymbolicModularEtaQuotient QMOD, List Matrix Integer) -> XHashTable(Matrix Integer, QEtaLaurentSeries CX)
- laurentExpansions(y, gammas)represents the- q-expansion of $- F_{- s,- r,- m,- t}(gammatau)$ for each gamma in the canonical variables given by the width of gamma- wrt. QMOD. It is an error if gammas are not a subset of transformationMatrices(- y).
- laurentExpansions: SymbolicModularEtaQuotient QMOD -> XHashTable(Matrix Integer, QEtaLaurentSeries CX)
- laurentExpansions yrepresents the- q-expansion of $- F_{- s,- r,- m,- t}(gammatau)$ at all cusps given by- yin the canonical variables given by the width of the cusp- wrt. QMOD.
- puiseuxExpansion: (SymbolicEtaQuotientLambdaGamma, Fraction Integer) -> QEtaPuiseuxSeries CX
- puiseuxExpansion(y, e)computes the Puiseux expansion of $- g_{- r,- m,lambda(gamma tau)$ in terms of $- q$ multiplied by $exp(2pi- i- e)$. The $(- ctau+d)$ factor is missing. See eqref{eq:g_r-- m-lambda(gamma*tau)}.
- puiseuxExpansion: SymbolicEtaQuotientGamma -> QEtaPuiseuxSeries CX
- puiseuxExpansion(y)computes the Puiseux expansion of the- q-expansion of $- p_{- r,- m,- t}(gamma tau)$, see eqref{eq:p_r-- m-- t(gamma*tau)}. The $(- ctau+d)$ factor is missing.
- puiseuxExpansion: SymbolicModularEtaQuotientGamma QMOD -> QEtaPuiseuxSeries CX
- puiseuxExpansion(y)represents the- q-expansion of $- F_{- s,- r,- m,- t}(gamma tau)$, see eqref{eq:F_s-- r-- m-- t(gamma*tau)}.
- qPochhammerPart: SymbolicEtaQuotientLambdaGamma -> QEtaPuiseuxSeries CX
- qPochhammerPart(y)returns the qPochhammer part of formula eqref{eq:g_r-- m-lambda(gamma*tau)}, i.e. a series of order 0 with constant term 1.