Let $\sldeuxr$ be the group of real $2\times 2$ matrices with determinant $1$ (see entry on special linear groups). The group $\sldeuxr$ acts on $H$ , the upper half plane, through fractional linear transformations. That is, if $$ \gamma = \begin{pmatrix}a & b \\ c & d\end{pmatrix}, $$ and $\tau\in H$ , then we let \begin{equation} \gamma \tau=\frac{a\tau+b}{c\tau+d}. \end{equation} For any natural number $N \geq 1$ , define the congruence subgroup $\Gamma_0(N)$ of level $N$ to be the following subgroup of the group $\sldeuxz$ of integer coefficient matrices of determinant $1$ : $$ \Gamma_0(N) := \left\{ \left. \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \sldeuxz\ \right|\ c \equiv 0 \pmod{N} \right\}. $$

Fix an integer $k$ . For $\gamma\in\sldeuxz$ and a function $f$ defined on $H$ , we define $$f_{\mid\gamma}(\tau)=\frac{f(\gamma \tau)}{(c\tau+d)^k}.$$ For a finite index subgroup $\Gamma$ of $\sldeuxz$ containing a congruence subgroup, a function $f$ defined on $H$ is said to be a weight $k$ modular form if:

1. $f=f_{\mid \gamma}$ for $\gamma \in \Gamma$ .
2. $f$ is holomorphic on $H$ .
3. $f$ is holomorphic at the cusps.

This last condition requires some explanation. First observe that the element $$ \mu = \begin{pmatrix} 1 & m \\ 0 & 1 \end{pmatrix} \in \Gamma_0(N), $$ and $\mu z = z + m$ , while if $f$ satisfies all the other conditions above, $f_{\mid \mu} = f$ . In other words, $f$ is periodic with period $1$ . Thus, convergence permitting, $f$ admits a Fourier expansion. Therefore, we say that $f$ is holomorphic at the cusps if, for all $\gamma \in \Gamma$ , $f_{\mid \gamma}$ admits a a Fourier expansion \begin{equation} f_{\mid \gamma}(\tau)=\sum_{n=0}^{\infty}a_{n}q^{n}, \end{equation}where $q=e^{2i\pi \tau}$ .

If all the $a_n$ are zero for $n\le 0$ , then a modular form $f$ is said to be a cusp form. The set of modular forms for $\Gamma$ (respectively cusp forms for $\Gamma$ ) is often denoted by $M_{k}(\Gamma)$ (respectively $S_{k}(\Gamma)$ ). Both $M_{k}(\Gamma)$ and $S_{k}(\Gamma)$ are finite dimensional vector spaces.

The space of modular forms for $\sldeuxz$ (respectively cusp forms) is non-trivial for any $k$ even and greater than 4 (respectively greater than $12$ and not $14$ ). Examples of modular forms for $\sldeuxz$ are:

1. The Eisenstein series $E_{m}$ , where $m$ is even and greater than $4$ , is a modular form of weight $m$ . Here $B_{m}$ denotes the $m$ -th Bernoulli number and, as usual, $q=e^{2i\pi \tau}$ : \begin{equation} E_{m}(\tau)=1-\frac{2m}{B_{m}}\underset{n=1}{\overset{\infty}{\sum}}\sigma_{m- 1}(n)q^n. \end{equation}For instance, \begin{equation} E_{4}(\tau)=1+240\underset{n=1}{\overset{\infty}{\sum}}\sigma_{3}(n)q^n \end{equation}and \begin{equation} E_{6}(\tau)=1-504\underset{n=1}{\overset{\infty}{\sum}}\sigma_{5}(n)q^n. \end{equation}
2. The Weierstrass $\Delta$ function, also called the modular discriminant, is a modular form of weight $12$ : \begin{equation} \Delta(\tau)=q\underset{n=1}{\overset{\infty}{\prod}}(1-q^n)^{24}. \end{equation}

Every modular form is expressible as \begin{equation} f(\tau)=\underset{n=0}{\overset{\lfloor{k/12}\rfloor}{\sum}}{a_n}{E_{k-12n}(\tau)}{(\Delta(\tau))^n}, \end{equation}where the $a_n$ are arbitrary constants, $E_0(\tau)=1$ and $E_2(\tau)=0$ . Cusp forms are the forms with $a_0=0$ .