Petersson scalar product (reviewed)

Let $f$ and $g$ be two modular forms with respect to a finite index subgroup $G$ of $\Gamma$. When it exists, we define the Petersson scalar product of $f$ and $g$ with respect to the group $G$ by \[ \langle f,g\rangle_G=\frac{1}{[\Gamma:G]}\int_{\mathfrak{F}}f(z)\overline{ g(z)}y^kd\mu, \] where $\mathfrak{F}$ is a fundamental domain for $G$ and $d\mu=dxdy/y^2$ is the measure associated to the hyperbolic metric.

Note that the Petersson scalar product exists if at least one of $f$, $g$ is a cusp form.