Old subspace of modular forms (reviewed)

Each space of $S_k(N,\chi)$ of cuspidal modular forms of weight $k$, level $N$, and character $\chi$ contains an old subspace $S_k^{\rm old}(N,\chi)$ that can be expressed as a direct sum of spaces of newforms $S_k^{\rm new}(N_i,\chi_i)$, where each $N_i$ is a proper divisor of $N$ divisible by the conductor of $\chi$, and each $\chi_i$ is the unique character of modulus $N_i$ induced by the primitive character that induces $\chi$.

This decomposition arises from the injective maps \[ \begin{aligned} \iota_d\colon S_k(N_i,\chi_i)&\to S_k(N,\chi)\\ f\ &\mapsto f(d\tau) \end{aligned} \] that exist for each divisor $d$ of $N/N_i$. The image of each $\iota_d$ is isomorphic to $S_k(N_i,\chi_i)$, and we have the decomposition \[ S_k(N,\chi)\simeq\!\!\!\! \bigoplus_{\mathrm{cond}(\chi)|N_i|N}\!\!\!\! S_k^{\rm new}(N_i,\chi_i)^{\oplus m_i}, \] where $m_i$ is the number of divisors of $N/N_i$. Restricting the direct sum to proper divisors $N_i$ of $N$ yields a decomposition for $S_k^{\rm old}(N,\chi)$.