$M_k(\Gamma_1(N)) = \bigoplus_{\chi \mod N} M_k(N, \chi)$ I am stuck with the following statement in the study of modular forms:
$$
M_k(\Gamma_1(N)) = \bigoplus_{\chi \mod N} M_k(N, \chi),
$$
where $\Gamma_1(N) := \left\{\begin{pmatrix}a & b\\
c & d\end{pmatrix} \in SL_2(\mathbb{Z}): a \equiv d \equiv 1 \mod N, \; N \mid c\right\}$ and $\chi$ runs over all Dirichlet characters modulo $N$. Any hints or proof directions?
 A: This requires some basic representation theory (or you can reproduce the representation-theoretic theorems in a special case).
Note that $\Gamma_1(N)$ is a normal subgroup of $\Gamma_0(N)$. So take any $\gamma \in \Gamma_0(N)$. Then because $\gamma \Gamma_1(N)=\Gamma_1(N)\gamma$ one can show that for $f \in M_k(\Gamma_1(N))$, we have $f|_k[\gamma] \in M_k(\Gamma_1(N))$. In this way, we obtain an action of $\Gamma_0(N)$ on $M_k(\Gamma_1(N))$. For this action, $\Gamma_1(N)$ acts trivially, so we obtain an action of the quotient group $\Gamma_0(N)/\Gamma_1(N)$. Now this quotient is actually isomorphic to $(\Bbb Z/N\Bbb Z)^\times$.
So we have an action $(\Bbb Z/N\Bbb Z)^\times$ on $M_k(\Gamma_1(N))$. Because the Petersson slash operator is $\Bbb C$-linear, we actually have a representation of the group $(\Bbb Z/N\Bbb Z)^\times$ on $M_k(\Gamma_1(N))$. By elementary representation theory, this representation decomposes into $\rho$-isotypic components, where $\rho$ runs over all irreducible representations of $(\Bbb Z/N\Bbb Z)^\times$. But irreducible representations of $(\Bbb Z/N\Bbb Z)^\times$ are exactly the Dirichlet characters and the isotypic component is just $M_k(N,\chi)$.

Bonus: all of the above works more generally in the case where $\Gamma$ and $G$ are two congruence subgroups such that $\Gamma$ is a normal subgroup of $G$. In this setting, let $g \in \Delta$, then we have $g \Gamma = \Gamma g$ by normality, so that we have $f|_k[g] \in M_k(\Gamma)$ for all $f \in M_k(\Gamma)$. So we obtain a linear action of $G$ on $M_k(\Gamma)$ for which $\Gamma$ acts trvially, thus we have a representation of the finite group $G/\Gamma$ on the vector space $M_k(\Gamma)$.
By representation theory, this decomposes as a direct sum of $\rho$-isotypic components, where $\rho$ runs over all irreducible representations of $G/\Gamma$. These $\rho$-isotypic components can be considered "modular forms of generalized Nebentypus". For example, $\Gamma(N)$ is normal in $\Gamma(1)$ and we have $\Gamma(1)/\Gamma(N) \cong \mathrm{SL}_2(\Bbb Z/N\Bbb Z)$, so $M_k(\Gamma(N))$ decomposes accordings to irreducible representations of $\mathrm{SL}_2(\Bbb Z/N\Bbb Z)$.
