SymbolicModularEtaQuotientGamma QMODΒΆ
qetasymbmod.spad line 132 [edit on github]
QMOD: QEtaModularCategory
SymbolicModularEtaQuotientGamma(QMOD) holds data to compute an eta-quotient expansions of $F_
{s
,r
,m
,t
}(gamma tau)$ for a particular matrix $gammainSL2Z$.. See eqref{eq:F_s-r
-m
-t
(gamma*tau)}.
- basefactor: % -> List SymbolicEtaQuotientGamma
basefactor(x)
returns the part of $F_
{s
,r
,m
,t
}(gamma tau)$ that is connected to the generating series of a(m*n+k) fork
in modularOrbit(rspec,m
,t
) (from QMOD).
- coerce: % -> OutputForm
from CoercibleTo OutputForm
- coerce: SymbolicEtaQuotientGamma -> %
coerce(y)
turns an element of SymbolicEtaQuotientGamma into this domain.
- cofactor: % -> SymbolicEtaQuotientGamma
cofactor(x)
returns the cofactor part to make $F_
{s
,r
,m
,t
}$ a modular function for $Gamma_0(N
)$ or $Gamma_1(N
)$ (depending on the parameter QMOD.
- etaQuotient: (QEtaSpecification, Matrix Integer) -> %
generalizedEtaQuotient(rspec, gamma) represents the expansion of $
g_r
(gamma tau)$ wherer
is given through rspec.
- etaQuotient: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger, Matrix Integer) -> %
etaQuotient(sspec, rspec, m, t, gamma)
represents the expansion of $F_
{s
,r
,m
,t
}(gamma tau)$ wheres
andr
are given throughsspec
andrspec
.
- etaQuotient: QEtaSpecification -> %
etaQuotient(rspec)
represents the expansion of $g_r
(tau)$ wherer
is given throughrspec
. It is the same as generalizedEtaQuotient(rspec
,matrix[[1,0],[0,1]]).
- hash: % -> SingleInteger
from Hashable
- hashUpdate!: (HashState, %) -> HashState
from Hashable
- latex: % -> String
from SetCategory
- minimalRootOfUnity: % -> PositiveInteger
minimalRootOfUnity(x)
returns thelcm
of minimalRootOfUnity(cofactor(x
)) andlcm
([minimalRootOfUnityb
forb
in basefactor(x
)]).
- one?: % -> Boolean
one?(x)
returnstrue
if the generalized eta-quotient corresponding tox
represents 1. This is the case if one?(first(basefactor(x
))) and one?(cofactor(x
)). If this function returnsfalse
, then it is not guaranteed thatx
does not represent the constant series 1; it might still be the case that basefactor(x
) represents the inverse of cofactor(x
).
- order: % -> Integer
order(x)
returns the order of thez
-expansion in terms of the z=q^(1/w) wherew
is the width of the cusp of the group corresponding to QMOD that corresponds to transformationMatrix(x
). If the input condition multiplier(first(basefactor(x
)))=1
is not met, the function aborts with an error.
- orderMin: % -> Fraction Integer
orderMin(x)
returns the expected order of thez
-expansion in terms of z=q^(1/w) wherew
is the width of the cusp of the group corresponding to QMOD that corresponds to transformationMatrix(x
). Note that this is a lower bound of the order. If u=ceiling(orderMin(x
)) then even this integer is a lower bound of the order of the expansion ofz
in terms of x=q^(1/w), since the coefficient corresponding to x^u may be zero. Further note that due to estimation it does not necessarily return an integer, but a rational number. Only for cases where multiplier(first(basefactor(x
)))=1
orderMin(x
) is equal to thetrue
order of the expansion.
- qExponent: % -> Fraction Integer
qExponent(x)
returns the order of theq
-expansion in terms of the originalq
. If the input condition multiplier(first(basefactor(x
)))=1
is not met, the function aborts with an error.
- qExponentMin: % -> Fraction Integer
qExponentMin(x)
returns the order of theq
-expansion in terms of the originalq
. Note that this exponent is only a lower bound for theq
-expansion. The coefficient corresponding to thisq
-power may be zero. Only for cases where multiplier(first(basefactor(x
)))=1
qExponentMin(x
) is equal to thetrue
order (inq
) of the expansion.