Polynomial interpolation of modular forms for Hecke groups

Extending work of J. Raleigh, we compute polynomials $P_{n,F}(x)$ associated to certain families $F=\{f_m{\}}_{m=3,4,\mathrm{.}\mathrm{.}\mathrm{.}}$ of modular forms for Hecke groups $G({\lambda }_m)$ with the property that $P_{n,F}(m)$ is the $n^{th}$ coefficient in the Fourier expansion of $f_m$. We express the $P_{n,F}$ in terms of the Fourier expansions of well-known Hauptmoduln, or in terms of certain divisor-sums. By studying the complex roots of the $P_n$, we relate them to Lehmer’s question about Ramanujan’s tau function. We review the theory of triangle functions and Hecke’s theory of modular forms in order to establish a basis for our code, some of which originates in the dissertation of J. Leo. The article is an account of numerical experiments; the only theorems in it belong to work by others that we review as described above.