Classical modular form (reviewed)

Let $k$ be a positive integer and let $\Gamma$ be a finite index subgroup of the modular group $\SL(2,\Z)$.

A (classical) modular form $f$ of weight $k$ on $\Gamma$, is a holomorphic function defined on the upper half plane $\mathcal{H}$, which satisfies the transformation property $f(\gamma z) = (cz+d)^k f(z)$ for all $z\in\mathcal{H}$ and $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in \Gamma$ and is holomorphic at all the cusps of $\Gamma$.

If $\Gamma$ contains the principal congruence subgroup $\Gamma(N)$ then $f$ is said to be a modular form of level $N$.

For each fixed choice of $k$ and $\Gamma$ the set of modular forms of weight $k$ on $G$ form a finite-dimensional $\mathbb{C}$-vector space denoted $M_k(\Gamma)$.

For the congruence subgroup $\Gamma_1(N)$ the space $M_k(\Gamma_1(N))$ decomposes as a direct sum of subspaces $M_k(N,\chi)$ over the group of Dirichlet characters $\chi$ of modulus $N$, where $M_k(N,\chi)$ is the subspace of forms $f\in M_k(N)$ that satisfy $f(\gamma z) = \chi(d)(cz+d)^k f(z)$ for all $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ in $\Gamma_0(N)$.

Elements of $M_k(N,\chi)$ are said to be modular forms of weight $k$, level $N$, and character $\chi$.

For trivial character $\chi$ of modulus $N$ we have $M_k(N,\chi)=M_k(\Gamma_0(N))$.