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

TL;DR

This paper demonstrates that for certain levels, all cusp forms of weight ≥4 can be expressed as products of Eisenstein series, enabling explicit Fourier expansion calculations at various cusps.

Contribution

It establishes a new representation of cusp forms as products of Eisenstein series for specific levels and weights, facilitating Fourier expansion computations.

Findings

Cusp forms at certain levels are linear combinations of Eisenstein series products.

In weight 2, forms obtained this way correspond to eigenforms with non-zero L-values.

Explicit Fourier expansions at arbitrary cusps are computed for examples.

Abstract

We show, for levels of the form $N = p^a q^b N'$ with $N'$ squarefree, that in weights $k \geq 4$ 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 $L(g, 1) \neq 0$. 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.