New subspaces of modular forms (awaiting review)

The space $M_k(N,\chi)$ of modular forms of level $N$, weight $k$, and character $\chi$ can be decomposed \[ M_k(N,\chi) = M_k^{\rm old}(N,\chi) \oplus M_k^{\rm new}(N,\chi) \] into old and new subspaces, defined using the corresponding decompositions of $E_k(N,\chi)$ and $S_k(N,\chi)$ as $M_k^{\rm old}(N,\chi) = E_k^{\rm old}(N,\chi) \oplus S_k^{\rm old}(N, \chi)$ and $M_k^{\rm new}(N,\chi) = E_k^{\rm new}(N,\chi) \oplus S_k^{\rm new}(N, \chi)$.