Character of a modular form (reviewed)

The character of an elliptic modular form $f$ of weight $k$ for the group $\Gamma$ is the Dirichlet character $\chi$ that appears in its transformation under the action of the defining group $\Gamma$. Namely, \[f(\gamma z) = \chi(d)(cz+d)^k f(z) \] for any $z\in\mathcal{H}$ and $\gamma = \begin{pmatrix} * & * \\ c & d \end{pmatrix}\in\Gamma$. Here $\Gamma$ is a subgroup of $\rm{SL}_2(\mathbb{Z})$ containing the principal congruence subgroup $\Gamma(N)$, and $\chi$ is a character mod $N$.