Central limit theorems for sums of quadratic characters, Hecke eigenforms, and elliptic curves

Abstract

We prove central limit theorems (under suitable growth conditions) for sums of quadratic characters, families of Hecke eigenforms of level 1 1 and weight k k , and families of elliptic curves, twisted by an L L -function satisfying certain properties. As a corollary, we obtain a central limit theorem for products χ ( p ) a f ( p ) \chi (p)a_f(p) where χ \chi is a quadratic Dirichlet character and f f is a normalized Hecke eigenform.