Weakly holomorphic modular forms for some moonshine groups

In an article in the Pure and Applied Mathematics Quarterly in 2008, Duke and Jenkins investigated a certain natural basis of the space of weakly holomorphic modular forms for the full modular group $S{L}_2(\mathbf{Z})$. We show here that their results can be generalized to certain moonshine groups, also allowing characters that are real on the underlying subgroup ${\mathrm{\Gamma }}_0(N)$.