Products of Eisenstein series and Fourier expansions of modular forms at cusps

Abstract

We show, for levels of the form $N=p^a{q}^b{N}^{\prime }$ with $N^{\prime }$ squarefree, that in weights $k\ge 4$ every cusp form $f\in {\mathcal{S}}_k(N)$ is a linear combination of products of certain Eisenstein series of lower weight. In weight $k=2$ we show that the forms $f$ which can be obtained in this way are precisely those in the subspace generated by eigenforms $g$ with $L(g,1)\ne 0$. As an application of such representations of modular forms we can calculate Fourier expansions of modular forms at arbitrary cusps and we give several examples of such expansions in the last section.

This is an update to a previous version of the article. In the new version we describe an algorithm (available at https://github.com/michaelneururer/products-of-eisenstein-series) that computes Fourier expansions at cusps

References in corpus (1)

• On the order of vanishing of newforms at cusps

Cited by in corpus (3)

• Fourier expansions at cusps
• Congruences satisfied by eta-quotients
• All modular forms of weight 2 can be expressed by Eisenstein series