On signs of Fourier coefficients of Hecke-Maass cusp forms on 𝐺𝐿₃

Abstract

We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group S L 3 ( Z ) \mathrm {SL}_3(\mathbb Z) . When the underlying form is self-dual, we show that there are ≫ ε X 5 / 6 − ε \gg _\varepsilon X^{5/6-\varepsilon } sign changes among the coefficients { A ( m , 1 ) } m ≤ X \{A(m,1)\}_{m\leq X} and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients A ( m , m ) A(m,m) for generic G L 3 \mathrm {GL}_3 cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of G L 3 \mathrm {GL}_3 cusp forms we establish. In addition, non-vanishing of the Fourier coefficients is studied under the Ramanujan-Petersson conjecture.