Splitting fields of X^n-X-1 and modular forms

In his article `On a theorem of Jordan', Serre considered the family of polynomials f_n(X) = X^n-X-1 and the counting function of the number of roots of f_n over the finite field F_p, seen as function in p. He explicitly showed the `modularity' of this function for n=3,4.
In this talk, I report on joint work with Alfio Fabio La Rosa and Chandrashekhar Khare, in which we treat the case n=5 in several different ways.