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 of correspondences in two different ways. generalization Atkin-Lehner involutions \Gamma_0(M) and its subgroup \Gamma_0^{(M')}(M) introduced multiplicities at cusps.