All modular forms of weight 2 can be expressed by Eisenstein series

TL;DR

This paper proves that all modular forms of weight greater than 1 can be represented as linear combinations of products of Eisenstein series, simplifying previous restrictions and refining the result for weights above 2.

Contribution

It demonstrates that all modular forms of weight >1 are expressible via products of Eisenstein series, removing previous obstructions and refining the representation for higher weights.

Findings

All modular forms of weight >1 can be expressed using Eisenstein series.

For weights >2, exactly two cusp expansions suffice.

The obstruction related to nonvanishing L-values is removed.

Abstract

We show that every elliptic modular form of integral weight greater than $1$ can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $\mathrm{L}$-values present in all previous work. For weights greater than $2$, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.