QEtaModularLambdaTools CΒΆ
qetamodlambdatool.spad line 198 [edit on github]
QEtaModularLambdaTools provides functions for working with the modular lambda function.
- applyLambdaTransformation: (Matrix Integer, Fraction SparseUnivariatePolynomial C) -> Fraction SparseUnivariatePolynomial C
By https://fungrim.org/entry/099301/ we can easily get the expansion of the modular lambda function at any cusp/transformation. applyLambdaTransformation(
g
,rf
) replaces the variable ofrf
by the respective rational function expression according to the transformation matrixg
.
- evalAtModularLambda: Fraction SparseUnivariatePolynomial C -> QEtaLaurentSeries C if C has Field
evalAtModularLambda(rf)
replaces the variable ofrf
with modularLambda() from the QFunctions package. The function caches its result.
- evalAtModularLambda: SparseUnivariatePolynomial C -> QEtaLaurentSeries C
evalAtModularLambda(pol)
replaces the variable ofpol
with modularLambda() from the QFunctions package. The function caches its results.
- transformAtModularLambda: (Fraction SparseUnivariatePolynomial C, Matrix Integer, Matrix Integer, Fraction Integer) -> QEtaLaurentSeries C if C has Field
transformAtModularLambda(r,g,t,w)
returns the series $r
(lambda(g
(t
tau)))$ expressed in the variable $q^
{frac{1}{w
}}$ where $q=
exp(pii
tau)$ and $t
$ is the transformation $taumapstofrac{atau}{d
}$ for some integers $a$ and $d
$.