A Sato–Tate law for GL (3)

Abstract

We consider statistical properties of Hecke eigenvalues $A_j(p,1)$ for fixed $p$ as ${\varphi }_j$ runs through a basis of Hecke–Maaß cusp forms for the group ${\mathrm{SL}}_3(\mathbb{Z})$. We show that almost all of them satisfy the Ramanujan conjecture at $p$ and that their distribution is governed by the Sato–Tate law.