  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.
