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

Convergence to the Plancherel measure of Hecke Eigenvalues