Convergence to the Plancherel measure of Hecke eigenvalues

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 $d$.