On Signs of Fourier Coefficients of Hecke-Maass Cusp Forms on ${\mathrm{G}\mathrm{L}}_3$

We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group ${\mathrm{S}\mathrm{L}}_3(\mathbb{Z})$. When the underlying form is self-dual, we show that there are ${\gg }_{\epsilon }{X}^{5\mathrm{/}6-\epsilon }$ sign changes among the coefficients $\{A(m,1){\}}_{m\le X}$ 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 $A(m,m)$ for generic ${\mathrm{G}\mathrm{L}}_3$ cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of ${\mathrm{G}\mathrm{L}}_3$ cusp forms we establish. In addition, non-vanishing of the Fourier coefficients is studied under the Ramanujan-Petersson conjecture.