  We give improved uniform estimates for the rate of convergence to Plancherel
measure of Hecke eigenvalues of holomorphic forms of weight 2 and level N.
These are applied to determine the sharp cutoff for the non-backtracking random
walk on arithmetic Ramanujan graphs and to Serre's problem of bounding the
multiplicities of modular forms whose coefficients lie in number fields of
degree d.
