Vanishing of Poincar\'e series for congruence subgroups

We consider the problem of the vanishing of Poincar\'e series for congruence subgroups. Denoting by Pk,m,N the Poincar\'e series of weight k and index m for the group Γ₀(N), we show that for certain choices of parameters $k,m,N$, the Poincar\'e series does not vanish. Our methods improve on previous results of Rankin (1980) and Mozzochi (1989).