Explicit zero-free regions for automorphic $L$-functions

Let $L(s,f)$ be the $L$-function associated with a newform $f$ of even weight $k$, squarefree level $N$ and trivial nebentypus. In this paper, we establish a new explicit zero-free region for $L(s,f)$. More precisely, we prove that $L(s,f)$ does not vanish in the region $\mathrm{\Re }(s)\ge 1-\frac{1}{C\log (kN\max (1,\mathrm{\mid }\mathrm{\Im }(s)\mathrm{\mid }))}$ with $C=16.7053$ if $\mathrm{\mid }\mathrm{\Im }(s)\mathrm{\mid }\ge 1$ or $\mathrm{\mid }\mathrm{\Im }(s)\mathrm{\mid }\le \frac{0.30992}{\log (kN)}$ and $C=16.9309$ if $\frac{0.30992}{\log (kN)}<\mathrm{\mid }\mathrm{\Im }(s)\mathrm{\mid }\le 1$. This improves a result of Hoey et al. where $445.994$ was shown to be an admissible value for $C$.