A new basis for the space of modular forms

Let $G_{2n}$ be the Eisenstein series of weight $2n$ for the full modular group $\mathrm{\Gamma }=S{L}_2(\text{ZZ})$. It is well-known that the space $M_{2k}$ of modular forms of weight $2k$ on $\mathrm{\Gamma }$ 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 $\{G_{2k}\}\cup \{G_{4i}{G}_{2k-4i}\mathrm{\mid }i=1,2,\dots ,d_k\}$ 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$d_k+1=\dim_{\CC} M_{2k}$.