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 of rf by the respective rational function expression according to the transformation matrix g.

evalAtModularLambda: Fraction SparseUnivariatePolynomial C -> QEtaLaurentSeries C if C has Field

evalAtModularLambda(rf) replaces the variable of rf with modularLambda() from the QFunctions package. The function caches its result.

evalAtModularLambda: SparseUnivariatePolynomial C -> QEtaLaurentSeries C

evalAtModularLambda(pol) replaces the variable of pol 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 $q=exp(pi i tau)$ and $t$ is the transformation $taumapstofrac{atau}{d}$ for some integers $a$ and $d$.