A Selberg-type zero-density result for twisted $\mathrm{G}{\mathrm{L}}_2$ $L$-functions and its application

Let $f$ be a fixed holomorphic primitive cusp form of even weight $k$, level $r$ and trivial nebentypus ${\chi }_r$. Let $q$ be an odd prime with $(q,r)=1$ and let $\chi$ be a primitive Dirichlet character modulus $q$ with $\chi \mathrm{\ne }{\chi }_r$. In this paper, we prove an unconditional Selberg-type zero-density estimate for the family of twisted $L$-functions $L(s,f\otimes \chi )$ in the critical strip. As an application, we establish an asymptotic formula for the even moments of the argument function $S(t,f\otimes \chi )={\pi }^{-1}\arg L(1\mathrm{/}2+\mathrm{ı}t,f\otimes \chi )$ and prove a central limit theorem for its distribution over $\chi$ of modulus $q$.