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.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by David Farmer on 2019-04-11 22:49:52

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**History:**(expand/hide all)

- 2019-04-11 22:49:52 by David Farmer (Reviewed)
- 2019-04-11 22:44:21 by David Farmer (Reviewed)
- 2016-03-25 15:16:01 by Andreea Mocanu

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