Space of newforms (reviewed)

For each level $N$, weight $k$, and character $\chi$ the space $S_k(N,\chi)$ of cuspidal modular forms can be decomposed as an internal direct sum \[ S_k(N,\chi) = S_k^{\rm old}(N,\chi) \oplus S_k^{\rm new}(N,\chi). \] The old subspace $S_k^{\rm old}(N,\chi)$ is generated by all elements of $S_k(N,\chi)$ of the form $f(m z)$ for $f\in S_k(M,\chi')$ where $\chi'$ induces $\chi$ and $m M | N$.

The new subspace $S_k^{\rm new}(\Gamma_0(N),\chi)$ is the orthogonal complement of $S_k^{\rm old}(N,\chi)$ with respect to the Petersson inner product on $S_k(N,\chi)$. The newforms in $S_k(N,\chi)$ are a canonical basis for this subspace.