Modular Forms in Pari/GP

TL;DR

This paper discusses the Pari/GP modular forms package, highlighting its capabilities in computing Fourier expansions, evaluating forms and L-functions, and calculating scalar products, thus advancing computational tools in modular forms.

Contribution

It introduces new computational features in the Pari/GP package, enabling more comprehensive analysis of modular forms and their associated functions.

Findings

First package to compute Fourier expansions at any cusp

Enables evaluation of modular forms near the real axis

Allows computation of L-functions of non-eigenforms

Abstract

We give theoretical and practical information on the Pari/GP modular forms package available since the spring of 2018. Thanks to the use of products of two Eisenstein series, this package is the first which can compute Fourier expansions at any cusps, evaluate modular forms near the real axis, evaluate L-functions of non-eigenforms, and compute general Petersson scalar products.