Kuznietsov trace formula and weighted distribution of Hecke eigenvalues

AbstractLet (X,μ) be a measurable topological space. Let S1,S2,… be a family of finite subsets of X. Suppose each x∈Si has a weight wix∈R+ assigned to it. We say {Si} is {wi}-distributed with respect to the measure μ if for any continuous function f on X, we have limi→∞∑x∈Siwixf(x)∑x∈Siwix=∫Xf(x)dμ(x).Let S(N,k) be the space of modular cusp forms over Γ0(N) of weight k and let E(N,k)⊂S(N,k) be a basis which consists of Hecke eigenforms. Let ar(h) be the rth Fourier coefficient of h. Let xph be the eigenvalue of h relative to the normalized Hecke operator T′p. Let ||·|| be the Petersson norm on S(N,k). In this paper we will show that for any even integer k⩾3, {xph:h∈E(N,k)},p∤N is {|ar(h)|2e−4πr||h||2}-distributed with respect to a polynomial times the Sato–Tate measure when N→∞