On the average least negative Hecke eigenvalue

The least quadratic non-residue has been a central problem in number theory for centuries. The average least quadratic non-residue was explored by Erdős in the 1960s, and many extensions of this problem such as to the average least character non-residue (Martin, Pollack) have been explored. In this talk, we look in to the average first sign change of Fourier coefficients of newforms (equivalently Hecke eigenvalues). We discuss the distribution of Hecke eigenvalues through the so-called 'horizontal' and 'vertical' Sato-Tate distributions, and we also discuss large sieve inequalities for cusp forms that are uniform in both the weight and the level.