be an approximation towards the Ramanujan–Petersson conjecture. Currently $\theta=1/9$ is known by the work of Kim and Shahidi [@KSh], while the Ramanujan–Petersson conjecture predicts $\theta=0$.

\[theorem1\] For any ${\varepsilon}>0$ one has $L(1/2,\pi\otimes \chi)
\ll_{\pi,\chi_\infty,K,{\varepsilon}} ({\mathcal{N}}{\mathfrak{q}})^{\frac{1}{2} -
\frac{1}{8}(1-2\theta)+{\varepsilon}}$.

This result contains a bound for all values $L(1/2+it,\pi\otimes\chi)$ on the critical line, because replacing $1/2$ by $1/2+it$ has the same effect as replacing $\chi$ by $\chi\otimes|\cdot|^{it}$.

The convexity bound in this context is $({\mathcal{N}}{\mathfrak{q}})^{\frac{1}{2}+{\varepsilon}}$. The first subconvex bound over totally number fields is a result by Cogdell, Piatetski-Shapiro and Sarnak [@CPSS; @Co], in which they obtained for $\pi$ induced by a holomorphic Hilbert cusp form[^2] $$L(1/2,\pi\otimes \chi) \ll ({\mathcal{N}}{\mathfrak{q}})^{\frac{1}{2} - \frac{1-2\theta}{14+4\theta} +{\varepsilon}}.$$ They used a very effective spectral method based on bounds for triple products [@Sa; @Sa2]. As an application of an ingenious and flexible geometric method, Venkatesh [@Ve Theorem 6.1] proved a subconvex bound over all number fields and for all irreducible cuspidal representations $$L(1/2,\pi\otimes \chi) \ll ({\mathcal{N}}{\mathfrak{q}})^{\frac{1}{2} - \frac{(1-2\theta)^2}{14-12\theta}+{\varepsilon}}.$$ Our method is quite different from both of these works, and Theorem \[theorem1\] supersedes both results. It may be noted that although most applications of subconvexity require only *any* nontrivial saving in the exponent, there are situations where the quality of 