# Decomposing the space of modular forms into $\chi$-eigenspaces via representation theory

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?

• $\Gamma_1(N)$ 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 $\chi$. – Furlo Roth Apr 12 at 19:06
• – Watson Apr 13 at 8:04