Hurwitz class numbers with level and modular correspondences

In this paper, we prove Hurwitz-Eichler type formulas for Hurwitz class numbers with each level $M$ when the modular curve $X_0(M)$ has genus zero. A key idea is to calculate intersection numbers of modular correspondences with the level in two different ways. A generalization of Atkin-Lehner involutions for $\Gamma_0(M)$ and its subgroup $\Gamma_0^{(M')}(M)$ is introduced to calculate intersection multiplicities of modular correspondences at cusps.