A new basis for the space of modular forms

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 $d_k+1=\dim_{\CC} M_{2k}$.