A Selberg-type zero-density result for twisted $\rm GL_2$ $L$-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 $\chi\neq\chi_r$. 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 $S(t, f \otimes \chi)=\pi^{-1}\arg L(1/2+\i t, f\otimes\chi)$ and prove a central limit theorem for its distribution over $\chi$ of modulus $q$.