A holomorphic function on $\mathbb{H}$ is called a modular form of weight $k$ for SL$_2(\mathbb{Z)}$ if it satisfies
- $f(Mz)= (cz+d)^kf(z) \ \ \ \ \ \ \ \forall M \in$ SL$_2(\mathbb{Z)}$
- $f$ is holomorphic at the cusp $\infty$
I know that its possible to consider congruence subgroups like $$\Gamma_0(m) = \{\begin{pmatrix}a & b \\ c & d\end{pmatrix} : c \equiv 0 \mod m \}$$ instead of SL$_2(\mathbb{Z)}$. But why do we have to consider more than one cusp in this case? I dont really understand it and i hope someone can explain this to me. Thanks in advance