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We study fluctuations in the distribution of families of   $p$  -th Fourier coefficients   $a_f(p)$   of normalized holomorphic Hecke eigenforms   $f$   of weight   $k$   with respect to   $SL_2({\mathbb Z})$   as   $k \to \infty$   and primes   $p \to \infty.$   These families are known to be equidistributed with respect to the Sato–Tate measure. We consider a fixed interval   $I \subset [-2,2]$   and derive the variance of the number of   $a_f(p)$  ’s lying in   $I$   as   $p \to \infty$   and   $k \to \infty$   (at a suitably fast rate). The number of   $a_f(p)$  ’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.

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