Cuspidal Modular Forms

Let $N$ be a positive integer and consider the set

$$\Gamma_0(N) = \left\{\left( \begin{matrix}a&b\\ c&d \end{matrix}\right)\in\SL_2(\mathbb{Z}) \,:\, N \mid c\right\}.$$

Definition 3.1 (Cuspidal Modular Form)   A cuspidal modular form of level $N$ is a holomorphic function $f:\mathfrak{h}\rightarrow \mathbb{C}$ such that

1. $f\vert _\gamma = f$ for all $\gamma\in\Gamma_0(N)$,
2. for every $\gamma\in\SL_2(\mathbb{Z})$,
   $$\lim_{z\rightarrow \infty} f(\gamma(z)) = 0,$$
   and
3. $f$ has a Fourier expansion:
   $$f = \sum_{n=1}^{\infty} a_n q^n.$$

Exercise 3.2   Prove that condition 3 is implied by conditions 1 and 2, so condition 3 is redundant. [Hint: Since $\gamma=\left( \begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}\right)\in\Gamma_0(N)$, condition 1 implies that $f(z+1)=f(z)$, so there is a function $F(q)$ on the open punctured unit disc such that $F(q(z)) = f(z)$. Condition 2 implies that $\lim_{q\rightarrow 0} F(q) = 0$, so by complex analysis $F$ extends to a holomorphic function on the full open unit disc.]

Definition 3.3   The $q$-expansion of $f$ is the Fourier expansion $f = \sum_{n=1}^\infty a_n q^n.$

Exercise 3.4   Suppose that $f\in S_2(\Gamma_0(N))$. Prove that

$$f(z) dz = f(\gamma(z))d(\gamma(z))$$

for all $\gamma\in\Gamma_0(N)$. [Hint: This is simple algebraic manipulation.]

Exercise 3.5   Let $S_2(\Gamma _0(N))$ denote the set of cuspidal modular forms of level $N$. Prove that $S_2(\Gamma _0(N))$ forms a $\mathbb{C}$-vector space under addition.