Asymptotics for Hecke eigenvalues with improved error term

Asymptotics for the distribution of Hecke eigenvalues in families of automorphic forms are useful to study families of L-functions, provided one has a sufficiently good estimate for the error term of the asymptotic. I want to present ongoing joint work with T. Finis in which we give an upper bound for the error term with an explicit dependence on the Hecke operator. For certain applications an improvement of this bound would be necessary, but to go beyond our bound it seems necessary to remove part of the geometric side of the trace formula.