SymbolicEtaGamma

qetasymb.spad line 172 [edit on github]

SymbolicEtaGamma collects data for the expansion of $eta_{delta,m,lambda}(gammatau)$ and $eta_{delta,g,m,lambda}^{[R]}(gammatau)$. See eqref{eq:eta_delta-m-lambda(gamma*tau)-YEG} and eqref{eq:eta_delta-g-m-lambda^[R](gamma*tau)-YEG}.

=: (%, %) -> Boolean

from BasicType

~=: (%, %) -> Boolean

from BasicType

cdExponent: % -> Fraction Integer

If pure?(x) cdExponent(x)=1/2, otherwise cdExponent(x)=0. The return value corresponds to the exponent of the (c*tau+d) factor. See eqref{eq:chi_delta-g}.

coerce: % -> OutputForm

from CoercibleTo OutputForm

delta: % -> PositiveInteger

If x=eta(delta,g,m,lambda,gamma) then delta(x) returns delta.

eta: (PositiveInteger, Integer, PositiveInteger, NonNegativeInteger, Matrix Integer) -> %

eta(delta,g,m,lambda,gamma) represents the meta-data for the expansion of $eta_{delta,m,lambda}}(gamma tau)$ and $eta_{delta,g,m,lambda}^{[R]}(gamma tau)$ in terms of $q=exp(2i pi tau)$. We require that c>0 in gamma=matrix[[a,b],[c,d]]. Data for a pure eta-quotient can be created setting g=-1.

gamma1: % -> Matrix Integer

If x=eta(delta,g,m,lambda,gamma) then gamma1(x) returns the SL2Z part of splitMatrix(gamma,delta,m,lambda).

hash: % -> SingleInteger

from Hashable

hashUpdate!: (HashState, %) -> HashState

from Hashable

lambda: % -> NonNegativeInteger

If x=eta(delta,g,m,lambda,gamma) then lambda(x) returns lambda.

latex: % -> String

from SetCategory

minimalRootOfUnity: % -> PositiveInteger

minimalRootOfUnity(x) returns the smallest positive integer n such that the expansion of the function $eta_{delta,m,lambda}(gamma tau)$ or $eta_{delta,g,m,lambda}^{[R]}(gamma tau)$ corresponding to x=eta(delta,g,m,lambda,gamma) (neglecting the (c*tau+d) factor) lives in Q[w][[z]] where w is an n-th root of unity and z a fractional q power.

minimalRootOfUnityWithoutPrefactors: % -> PositiveInteger

minimalRootOfUnityWithoutPrefactos(x) returns the smallest positive integer n such that the expansion of the function $eta_{delta,m,lambda}(gamma tau)$ or $eta_{delta,g,m,lambda}^{[R]}(gamma tau)$ corresponding to x=eta(delta,g,m,lambda,gamma) (neglecting the (c*tau+d) factor, the square root of the rational prefactor (rationalPrefactor(x), and the denominator of v_{delta,g,m,lambda,gamma} from eqref{v_delta-g-m-lambda-gamma}, i.e. unityExponent(x)) lives in Q[w][[z]] where w is an n-th root of unity and z a fractional q power.

multiplier: % -> PositiveInteger

If x=eta(delta,g,m,lambda,gamma) then multiplier(x) returns m.

pure?: % -> Boolean

pure?(x) returns true if x corresponds to a pure eta function.

qExponent: % -> Fraction Integer

qExponent(x) returns the (fractional) exponent $u_{delta,g,m,lambda,gamma}$ for the order of the expansion of x in the original variable q, see eqref{eq:u_delta-g-m-lambda-gamma}.

rationalPrefactor: % -> Fraction Integer

rationalPrefactor(x) returns the square of the second product in eqref{eq:eta_delta-m-lambda(gamma*tau)} if pure?(x). See ref{eq:rho_delta-g-m-lambda}. If not pure?(x) then rationalPrefactor(x)=1.

subindex: % -> Integer

If x=eta(delta,g,m,lambda,gamma) then subindex(x) returns g.

transformationMatrix: % -> Matrix Integer

If x=eta(delta,g,m,lambda,gamma) then transformationMatrix(x) returns gamma.

udelta: % -> Fraction Integer

Returns $u_{delta,m,lambda}$. See eqref{eq:uv_delta}

unityExponent: % -> Fraction Integer

unityExponent(x) returns the fractional part of the exponent $v_{delta,g,m,lambda,gamma}$ for the unity factor of the expansion of x, see eqref{eq:v_delta-g-m-lambda-gamma}.

vdelta: % -> Fraction Integer

Returns $v_{delta,m,lambda}$. See eqref{eq:uv_delta}.

BasicType

CoercibleTo OutputForm

Hashable

SetCategory