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.

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- Last edited by David Roe on 2018-10-12 19:32:03

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- 2018-10-12 19:32:03 by David Roe (Reviewed)