SymbolicEtaGammaΒΆ
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}.
- 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 thatc>0in gamma=matrix[[a,b],[c,d]]. Data for a pure eta-quotient can be created settingg=-1.
- gamma1: % -> Matrix Integer
If x=eta(delta,
g,m,lambda,gamma) thengamma1(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 integernsuch 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 inQ[w][[z]] wherewis ann-th root of unity andza fractionalqpower.
- minimalRootOfUnityWithoutPrefactors: % -> PositiveInteger
minimalRootOfUnityWithoutPrefactos(
x) returns the smallest positive integernsuch 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 ofv_{delta,g,m,lambda,gamma} from eqref{v_delta-g-m-lambda-gamma}, i.e. unityExponent(x)) lives inQ[w][[z]] wherewis ann-th root of unity andza fractionalqpower.
- multiplier: % -> PositiveInteger
If x=eta(delta,
g,m,lambda,gamma) then multiplier(x) returnsm.
- pure?: % -> Boolean
pure?(x)returnstrueifxcorresponds 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 ofxin the original variableq, 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) returnsg.
- transformationMatrix: % -> Matrix Integer
If x=eta(
delta,g,m,lambda,gamma) then transformationMatrix(x) returns gamma.