Fluctuations in the Distribution of Hecke Eigenvalues about the Sato-Tate Measure

Abstract

We study fluctuations in the distribution of families of p-th Fourier coefficients af(p) of normalized holomorphic Hecke eigenforms f of weight k with respect to SL2(Z) as k→∞ and primes p→∞. These families are known to be equidistributed with respect to the Sato–Tate measure. We consider a fixed interval I⊂[−2,2] and derive the variance of the number of af(p)’s lying in I as p→∞ and k→∞ (at a suitably fast rate). The number of af(p)’s lying in I is shown to asymptotically follow a Gaussian distribution when appropriately normalized. A similar theorem is obtained for primitive Maass cusp forms