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 $Γ=SL_2(\ZZ)$. It is well-known that the space $M_{2k}$ of modular forms of weight $2k$ on $\Gamma$ has a basis $\{G_{4}^αG_{6}^β |\ α,β\in\ZZ,\ α,β\geq 0,\ 4α+6β=2k\}$. In this paper we will exhibit another (simpler) basis for $M_{2k}$. It is given by $\{G_{2k}\}\cup \{G_{4i}{G}_{2k-4i}|i=1,2,\dots ,d_k\}$ if $2k\equiv 0(\mathrm{mod}4)$, and $\{G_{2k}\}\cup \{G_{4i+2}{G}_{2k-4i-2}|i=1,2,\dots ,d_k\}$ if $2k\equiv 2(\mathrm{mod}4)$ where $d_k+1=\dim_{\CC} M_{2k}$.

AMS-LaTeX, 6 pages