Let $p$ be a prime for which the congruence group $\Gamma_0(p)^*$ is of genus zero, and let $j_p^*$ be the corresponding Hauptmodul. Let $f$ be a nearly holomorphic modular form of weight 1/2 on $\Gamma_0(4p)$ which satisfies some congruence condition on its Fourier coefficients. We interpret $f$ as a vector valued modular form. Applying Borcherds lifting of vector valued modular forms we construct infinite products associated to $j_p^*$ and relate them to Zagier's trace formula for the singular values of $j_p^*$.