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

Question

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?