Petersson inner product

《A First Course in Modular Forms》 · 第 182 页 · Section 5.4「The Petersson inner product」，Definition 5.4.1

Let $\Gamma\subset\mathrm{SL}_2(\mathbb Z)$ be a congruence subgroup. The Petersson inner product $\langle\ ,\ \rangle_\Gamma:S_k(\Gamma)\times S_k(\Gamma)\to\mathbb C$ is given by $\langle f,g\rangle_\Gamma=V_\Gamma^{-1}\int_{X(\Gamma)}f(\tau)\overline{g(\tau)}(\operatorname{Im}\tau)^k\,d\mu(\tau)$.