Subspaces of modular forms (reviewed)

Each space of modular forms $M_k(N,\chi)$ of level $N$, weight $k$, and character $\chi$ can be decomposed into its subspaces of Eisenstein series $E_k(N,\chi)$ and cusp forms $S_k(N,\chi)$, each of which can be further decomposed into old and new spaces.