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
)$QMOD forc
inspitzen
].
- etaQuotientTraceExpansions: (QEtaSpecification, PositiveInteger, List Matrix Integer) -> XHashTable(Matrix Integer, QEtaLaurentSeries CX)
etaQuotientTraceExpansions(spec,mm,gammas)
returns expansions of the (modularwrt
. QMOD corresponding tonn
) eta-quotient given byspec
at each gamma ofgammas
where nn=level(spec
). See definition of the trace operator in cite{Kohnen_WeierstrassPointsAtInfinity_2004
}. Note that if level(spec
)=mm
then etaQuotientTraceExpansions(spec
,mm
,gammas
) is equal to laurentExpansions(y
,gammas
) for y=etaQuotient(spec
,gammas
)$SymbolicModularEtaQuotient(QMOD).
- laurentExpansion: SymbolicModularEtaQuotientGamma QMOD -> QEtaLaurentSeries CX
laurentExpansion(y)
represents theq
-expansion of $F_
{s
,r
,m
,t
}(gammatau)$ given byy
in the canonical variables given by the width of the cuspwrt
. QMOD.
- laurentExpansionInfinity: SymbolicModularEtaQuotientGamma QMOD -> QEtaLaurentSeries CX
laurentExpansionInfinity(y)
represents theq
-expansion of $F_
{s
,r
,m
,t
}(tau)$ given byy
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 forc
in spitzen].
- laurentExpansions: (SymbolicModularEtaQuotient QMOD, List Matrix Integer) -> XHashTable(Matrix Integer, QEtaLaurentSeries CX)
laurentExpansions(y, gammas)
represents theq
-expansion of $F_
{s
,r
,m
,t
}(gammatau)$ for each gamma in the canonical variables given by the width of gammawrt
. QMOD. It is an error if gammas are not a subset of transformationMatrices(y
).
- laurentExpansions: SymbolicModularEtaQuotient QMOD -> XHashTable(Matrix Integer, QEtaLaurentSeries CX)
laurentExpansions y
represents theq
-expansion of $F_
{s
,r
,m
,t
}(gammatau)$ at all cusps given byy
in the canonical variables given by the width of the cuspwrt
. 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 $exp(2pii
e
)$. The $(c
tau+d)$ factor is missing. See eqref{eq:g_r-m
-lambda(gamma*tau)}.
- puiseuxExpansion: SymbolicEtaQuotientGamma -> QEtaPuiseuxSeries CX
puiseuxExpansion(y)
computes the Puiseux expansion of theq
-expansion of $p_
{r
,m
,t
}(gamma tau)$, see eqref{eq:p_r-m
-t
(gamma*tau)}. The $(c
tau+d)$ factor is missing.
- puiseuxExpansion: SymbolicModularEtaQuotientGamma QMOD -> QEtaPuiseuxSeries CX
puiseuxExpansion(y)
represents theq
-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.