SymbolicEtaQuotient

qetaquotsymb.spad line 976 [edit on github]

SymbolicEtaQuotient holds data to compute an eta-quotient expansions of g_r(tau) at all cusps of $Gamma_0(N)$ or at the given cusps. See eqref{g_r(tau)}

=: (%, %) -> Boolean

from BasicType

~=: (%, %) -> Boolean

from BasicType

coerce: % -> OutputForm

from CoercibleTo OutputForm

cusps: % -> List Cusp

If x=etaQuotient(mm, r), then cusps(x)=cusp(mm)$CongruenceSubgroupGamma0.

elt: (%, Cusp) -> SymbolicEtaQuotientGamma

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

etaQuotient: (PositiveInteger, List Integer) -> %

etaQuotient(mm, r) represents the expansion of g_r(tau) at all cusps a/c of Gamma0(mm) in terms of x = exp(2 pi i tau/w) where w=width(m, c) and gamma=cuspToMatrix(mm, a/c). It is the same as etaQuotient(mm, r, cusps(mm)$CongruenceSubgroupGamma0). $r$ is indexed by the divisors of $mm$. See eqref{eq:g_r(tau)}.

etaQuotient: (PositiveInteger, List Integer, List Cusp) -> %

etaQuotient(mm, r, cusps) represents the expansion of g_r(tau) at the given cusps a/c of Gamma0(mm) in terms of x = exp(2 pi i tau/w) where w=width(m, c) and gamma=cuspToMatrix(m, a/c). $r$ is indexed by the divisors of $mm$. See eqref{eq:g_r(tau)}.

etaQuotient: (QEtaSpecification, List Cusp) -> %

etaQuotient(rspec, cusps) represents the expansion of g_r(tau) at the given cusps a/c of Gamma0(mm) in terms of x = exp(2 pi i tau/w) where w=width(m, c) and gamma=cuspToMatrix(m, a/c). $r$ and $mm$ are given by rspec. See eqref{eq:g_r(tau)}.

etaQuotient: QEtaSpecification -> %

etaQuotient(rspec) represents the expansion of g_r(tau) at all cusps a/c of Gamma0(mm) in terms of x = exp(2 pi i tau/w) where w=width(m, c) and gamma=cuspToMatrix(mm, a/c). It is the same as etaQuotient(rspec, cusps(mm)$CongruenceSubgroupGamma0). $r$ and $mm$ are given by rspec. See eqref{eq:g_r(tau)}.

exponents: % -> List Integer

If x=etaQuotient(m, r) then exponents(x) returns the list of exponents, i.e. exponents(x)=r.

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

latex: % -> String

from SetCategory

level: % -> PositiveInteger

If x=etaQuotient(mm, r) then level(x) returns the level of the eta-quotient, i.e. level(x)=mm.

minimalRootOfUnity: % -> PositiveInteger

If x=etaQuotient(mm, r), then minimalRootOfUnity(x) returns the smallest positive integer n such that the expansion of the function g_r(tau) at any cusp of Gamma_0(mm) (neglecting the (ctau+d)^* factor lives in Q[w][[x]] where w is an n-th root of unity.

one?: % -> Boolean

one?(x) returns true if the eta-quotient corresponding to x represents 1.

qetaGrades: % -> List Integer

qetaGrades(x) returns the (negated) exponents of a series expansion of x at all the cusps in the canonical variables, i.e. in x=q^(1/w) where w=width(level x, denom cusp)$GAMMA0 Internally, the matrix of Ligozat is employed.

specification: % -> QEtaSpecification

If x=etaQuotient(spec,…), then specification(x) returns spec.

BasicType

CoercibleTo OutputForm

SetCategory