I'm reading Diamond and Shurman's introductory book on modular forms and, in chapter 4.3, they give a decomposition of $M_k(\Gamma_1(N))$ as a direct sum of eigenspaces defined for Dirichlet characters. Specifically, if $\chi$ is a Dirichlet character mod $N$ they define the $\chi$-eigenspace of $M_k(\Gamma_1(N))$ by $$ M_k(N,\chi)=\{f\in M_k(\Gamma_1(N))\mid f[\gamma]_k=\chi(d_\gamma),\, \,\, \forall \gamma\in \Gamma_0(N)\}, $$ where $d_\gamma$ is the lower right entry of $\gamma$. They then state that $M_k(\Gamma_1(N))=\oplus_\chi M_k(N,\chi)$ and mention (as a hint in the back of the book) that this decomposition follows from "a standard result from representation theory". What result are they referring to here? In other words, how would I set up this problem (proving that $M_k(\Gamma_1(N))=\oplus_\chi M_k(N,\chi)$) in the context of representation theory, and what 'basic' result am I meant to apply?

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    $\begingroup$ $\Gamma_1(N)$ is a normal subgroup of $\Gamma_0(N)$. It follows by construction that $M_k(\Gamma_1(N))$ is a module over the group $\Delta = \Gamma_0(N)/\Gamma_1(N)$. The "basic" result is now that any representation (over $\mathbf{C}$) of any finite abelian group $\Delta$ can be written as a direct sum of $1$-dimensional representations of $\Delta$. The one dimensional representations of $\Delta$ are just given by characters $\chi$ on which $\Delta$ acts via $\chi$. $\endgroup$ – Furlo Roth Apr 12 at 19:06
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    $\begingroup$ See Representation theory of finite abelian groups - Garrett. $\endgroup$ – Watson Apr 13 at 8:04

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