We exhibit a numerical method to compute a three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups. As one application, we find an explicit rational function of degree 50 which regularly realizes the group $PSU_3(5)$ as a Galois group over the rationals. This is joint work with Michael Klug, Michael Musty, and Sam Schiavone.