A new basis for the space of modular forms

TL;DR

This paper introduces a simpler basis for the space of modular forms of even weight on the full modular group, using Eisenstein series, which simplifies the understanding and computation of these spaces.

Contribution

The paper presents a new, simpler basis for the space of modular forms, replacing the classical basis with one involving Eisenstein series of specific weights.

Findings

New basis expressed via Eisenstein series of specific weights

Simplifies the structure and computation of modular form spaces

Applicable to all even weights on the full modular group

Abstract

Let $G_{2n}$ be the Eisenstein series of weight $2n$ for the full modular group $\Gamma=SL_2(\ZZ)$. It is well-known that the space $M_{2k}$ of modular forms of weight $2k$ on $\Gamma$ has a basis $\{G_{4}^\alpha G_{6}^\beta\ |\ \alpha,\beta\in\ZZ,\ \alpha,\beta\geq 0,\ 4\alpha+6\beta=2k\}$. In this paper we will exhibit another (simpler) basis for $M_{2k}$. It is given by $\{G_{2k}\}\cup\{G_{4i}G_{2k-4i}\ |\ i=1,2,\ldots,d_k\}$ if $2k\equiv 0\pmod 4$, and $\{G_{2k}\}\cup\{G_{4i+2}G_{2k-4i-2}\ |\ i=1,2,\ldots,d_k\}$ if $2k\equiv 2\pmod 4$ where $d_k+1=\dim_{\CC} M_{2k}$.