  After Zagier proved that the traces of singular moduli $j(z)$ are Fourier
coefficients of a weakly holomorphic modular form, various properties of the
traces of the singular values of modular functions mostly on the full modular
group $PSL_2(\mathbb{Z})$ have been investigated such as their exact formulas,
limiting distribution, duality, and congruences. The purpose of this paper is
to generalize these arithmetic properties of traces of singular values of a
weakly holomorphic modular function on the full modular group to those on a
congruence subgroup $\Gamma_0(N)$.
