A ${\mathrm{G}\mathrm{L}}_3$ level analog of Selberg's result for $S(t)$

Let $\pi$ be a spherical cuspidal ${\mathrm{G}\mathrm{L}}_3$ newform of prime level $N$ and trivial central character, and define $S(t,\pi )={\pi }^{-1}\arg L(1\mathrm{/}2+\text{i}t,\pi )$. Assuming the generalized Riemann hypothesis for $L(s,\pi )$, we establish an asymptotic formula for each fixed weighted moment of $S(t,\pi )$ as $N\to \mathrm{\infty }$, for fixed $t>0$. Consequently, under the same weights, $S(t,\pi )\mathrm{/}\sqrt{\log \log N}$ converges in distribution to a centered Gaussian distribution of variance $1\mathrm{/}(2{\pi }^2)$. The key input is a weighted orthogonality formula derived from the Kuznetsov formula of prime level.