SymbolicEtaQuotientQSeriesTools(C, xiord, CX, xi, MODG)¶

qetasymbeqqseriestool.spad line 128 [edit on github]

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 gammas:=[cuspToMatrix(c)$MODG for c in spitzen].

etaQuotientTraceExpansions: (QEtaSpecification, PositiveInteger, List Matrix Integer) -> XHashTable(Matrix Integer, QEtaLaurentSeries CX): etaQuotientTraceExpansions(spec,mm,gammas) returns expansions of the (modular wrt. the group represented by MODG) eta-quotient given by spec at each gamma of gammas. See definition of the trace operator in cite{Kohnen_WeierstrassPointsAtInfinity_2004}. Note that if groupLevel()$MODG=mm then etaQuotientTraceExpansions(spec,mm,gammas) is equal to laurentExpansions(y,gammas) for y=etaQuotient(spec,gammas)$SymbolicModularEtaQuotient(MODG). mm is supposed to be a divisor of groupLevel()$MODG.

laurentExpansion: SymbolicModularEtaQuotientGamma MODG -> QEtaLaurentSeries CX: laurentExpansion(y) represents the q-expansion of $F_{s,r,m,t}(gammatau)$ given by y in the canonical variables given by the width of the cusp wrt. MODG.

laurentExpansionInfinity: SymbolicModularEtaQuotientGamma MODG -> QEtaLaurentSeries CX: laurentExpansionInfinity(y) represents the q-expansion of $F_{s,r,m,t}(tau)$ given by y wrt. MODG. See eqref{eq:F_s-r-m-t(tau)}.

laurentExpansions: (SymbolicModularEtaQuotient MODG, List Cusp) -> XHashTable(Matrix Integer, QEtaLaurentSeries CX): laurentExpansions(y, spitzen) returns laurentExpansions(y,gammas) for nn:=level(y) and gammas:=[cuspToMatrix(c)$MODG for c in spitzen].

laurentExpansions: (SymbolicModularEtaQuotient MODG, 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. MODG. It is an error if gammas are not a subset of transformationMatrices(y).

laurentExpansions: SymbolicModularEtaQuotient MODG -> XHashTable(Matrix Integer, QEtaLaurentSeries CX): laurentExpansions y represents the q-expansion of $F_{s,r,m,t}(gammatau)$ at all cusps given by y in the canonical variables given by the width of the cusp wrt. MODG.

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 MODG -> 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.