  For each of the groups PSL2(F25), PSL2(F32), PSL2(F49), PGL2(F25), and
PGL2(F27), we display the first explicitly known polynomials over Q having that
group as Galois group. Each polynomial is related to a Galois representation
associated to a modular form. We indicate how computations with modular Galois
representations were used to obtain these polynomials. For each polynomial, we
also indicate how to use Serre's conjectures to determine the modular form
giving rise to the related Galois representation.
