q-expansion of a modular form (reviewed)

The q-expansion of a modular form $f(z)$ is its Fourier expansion at the cusp $z=i\infty$ ($q=0$), expressed as a power series $\sum a_n q^n$ in the variable $q=e^{2\pi iz}$.

For cusp forms the constant coefficient $a_0$ of the $q$-expansion is zero.