SymbolicSiftedEtaQuotientLambdaGamma

qetafun.spad line 2412

SymbolicSiftedEtaQuotientLambdaGamma is a generalization of SymbolicEtaQuotientGamma. It holds data to compute an eta quotient expansion of g_{s,m,t,lambda}(gamma tau) or of g_{s,m,-,lambda}(gamma tau). See eqref{eq:g_s-m-t-lambda(gamma*tau)} and eqref{eq:g_s-m—lambda(gamma*tau)}.

=: (%, %) -> Boolean

from BasicType

~=: (%, %) -> Boolean

from BasicType

coerce: % -> OutputForm

from CoercibleTo OutputForm

divisors: % -> List PositiveInteger

divisors(x) returns the divisors of level(x) that were given at creation time of x.

elt: (%, PositiveInteger ) -> SymbolicSiftedEtaDeltaLambdaGamma

x.delta returns the data corresponding to the respective delta.

etaQuotient: (PositiveInteger , List PositiveInteger , List Integer , PositiveInteger , Integer , NonNegativeInteger , Matrix Integer ) -> %

etaQuotient(mm, divs, s, m, t, lambda, gamma) represents the expansion of $g_{s,m,t,lambda}(gamma tau)$ in terms of $x = exp(2 pi i tau/w)$ where w=width(nn, c) and gamma=cuspToMatrix(nn, a/c) for a cusp a/c.

exponents: % -> List Integer

exponents(x) returns the list of exponents corresponding to all divisors.

gamma: % -> Matrix Integer

gamma(x) returns the transformation corresponding to x.

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState , %) -> HashState

from SetCategory

lambda: % -> NonNegativeInteger

lambda(x) returns lambda

latex: % -> String

from SetCategory

level: % -> PositiveInteger

level(x) returns the level of the eta quotient.

minimalRootOfUnity: % -> PositiveInteger

minimalRootOfUnity(e) returns the smallest positive integer n such that the expansion of the function $g_{s,m,t,lambda}(gamma tau)$ corresponding to e=etaQuotient(mm, divs, s, m, t, lambda, gamma) (neglecting the (c tau+d)^* factor) lives in Q[w][[z]] where w is a n-th root of unity and z a fractional q power.

multiplier: % -> PositiveInteger

multiplier(x) returns the subsequence multiplier. Returns m.

offset: % -> Integer

offset(x) returns the subsequence offset. Returns t.

qExponent: % -> Fraction Integer

qExponent(e) returns 24 times the order of the expansion of e in q = exp(2 pi i tau) while neglecting the (ctau+d) factor. It corresponds to 24 times the exponent of the fourth product of eqref{eq:g_s-m-t-lambda(gamma*tau)}.

rationalPrefactor: % -> Fraction Integer

rationalPrefactor(x) returns the square of the second product in eqref{eq:g_s-m-t-lambda(gamma*tau)}.

unityPower: % -> Fraction Integer

unityPower(e) returns $- lambda*(24*t+sumdelta{s})/(24*m) + sum_delta sdelta*(vdelta+Kappa_delta)/24$. It corresponds to the second factor of eqref{eq:g_s-m-t-lambda(gamma*tau)}.

BasicType

CoercibleTo OutputForm

SetCategory