Decomposition into newforms for Gamma1(N) (reviewed)

The space of newforms of weight \(k\) on \(\Gamma_1(N)\) decomposes as \[ S_k^{\mathrm{new}}(\Gamma_1(N)) \cong \bigoplus_{\chi \bmod N} S_k^{\mathrm{new}}(N, \chi), \] where the direct sum is over all Dirichlet characters modulo \(N\). If \(\chi\) and \(\chi'\) are in the same Galois orbit, then \(S_k^{\mathrm{new}}(N, \chi)\) and \(S_k^{\mathrm{new}}(N, \chi')\) are Galois conjugate and we identify them since we define newforms as Galois orbits.