Eisenstein series identities based on partial fraction decomposition

Abstract

From the theory of modular forms, we know that 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 determine a basis for the space of modular forms of weight $$k$$ on $$\mathrm{SL}_2 ({\mathbb Z})$$ .