Fourier coefficients of a modular form (reviewed)

Let \(f\) be a modular form on a finite index subgroup \(\Gamma\) of $\SL_2(\Z)$, and suppose $\Gamma$ contains the matrix $T:=\left(\begin{matrix}1&1\\0&1\end{matrix}\right)$. Then $f$ is periodic with period 1, so it has a Fourier expansion of the form \[ f(z)=\sum_{n\ge 0}a_n q^n , \] where $q=e^{2 \pi i z}$. That is the Fourier expansion of $f$ around the cusp $\infty$, with Fourier coefficients $a_n$. If one says "the Fourier expansion of $f$", is it understood to refer to the expansion at $\infty$.

For other cusps of $\Gamma$, suppose $w$ is the width of the cusp $\gamma\infty$, for some cusp representative $\gamma$. Then we can write $f$ as $f(z)=g_\gamma(e^{2\pi iz/w})$ for some holomorphic function $g_\gamma$ on the punctured unit disk. We can expand $g$ as a Laurent series: \[ g_\gamma(q^{1/w})=\sum_{n\geq0}a_\gamma(n)q^{n/w}\quad\text{for}\quad0<|q|<1. \]

We then define the Fourier expansion of $f$ around the cusp $\gamma\infty$ to be \[ f(z)=\sum_{n \geq0}a_\gamma(n)q^{n/w}, \] where $q=e^{2\pi iz}$.

The $a_\gamma(n)$ are called the Fourier coefficients of $f$ with respect to the cusp $\gamma\infty$.