  From the theory of modular forms, there are exactly $[(k-2)/6]$ linear
relations among the Eisenstein series $E_k$ and its products $E_{2i}E_{k-2i}\
(2\le i \le [k/4])$. We present explicit formulas among these modular forms
based on the partial fraction decomposition, and use them to determining a
basis of the space of modular forms of weight $k$ on ${\rm SL}_2({\mathbb Z})$.
