SymbolicModularEtaQuotientGamma QMODΒΆ

qetasymbmod.spad line 132 [edit on github]

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)}.

=: (%, %) -> Boolean

from BasicType

~=: (%, %) -> Boolean

from BasicType

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) for k 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 $g_r(gamma tau)$ where r is given through rspec.

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

etaQuotient(sspec, rspec, m, t, gamma) represents the expansion of $F_{s,r,m,t}(gamma tau)$ where s and r are given through sspec and rspec.

etaQuotient: QEtaSpecification -> %

etaQuotient(rspec) represents the expansion of $g_r(tau)$ where r is given through rspec. 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 the lcm of minimalRootOfUnity(cofactor(x)) and lcm([minimalRootOfUnity b for b in basefactor(x)]).

one?: % -> Boolean

one?(x) returns true if the generalized eta-quotient corresponding to x represents 1. This is the case if one?(first(basefactor(x))) and one?(cofactor(x)). If this function returns false, then it is not guaranteed that x 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 the z-expansion in terms of the z=q^(1/w) where w 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 the z-expansion in terms of z=q^(1/w) where w 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 of z 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 the true order of the expansion.

qExponent: % -> Fraction Integer

qExponent(x) returns the order of the q-expansion in terms of the original q. 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 the q-expansion in terms of the original q. Note that this exponent is only a lower bound for the q-expansion. The coefficient corresponding to this q-power may be zero. Only for cases where multiplier(first(basefactor(x)))=1 qExponentMin(x) is equal to the true order (in q) of the expansion.

BasicType

CoercibleTo OutputForm

Hashable

SetCategory