Oldspace decomposition for newform space (reviewed)

There are natural inclusion maps from lower level; the old subspace is the span of their images. For \(M\) dividing the level \(N\) and divisible by \(\operatorname{cond}(\chi)\), and for \(t\) dividing \(\frac{N}{M}\), the maps $$ \begin{aligned} \alpha_{M, t} : S_k(\Gamma_0(M), \chi) & \to S_k(\Gamma_0(N), \chi) \\\\ f(q) & \mapsto f(q^t) \end{aligned} $$ induce an isomorphism \[ S_k(\Gamma_0(N), \chi) \cong \bigoplus_{M,t} \alpha_{M,t}\left(S_k^{\mathrm{new}}(\Gamma_0(M), \chi)\right). \]