a typical off-diagonal term of the second moment. In this paper we generalize the method of [@BH] to (congruence subgroups of) the Hilbert modular group of a totally real number field $K$, and we give applications to subconvexity of twisted $L$-functions over $K$. Before stating our main result, we note that the subconvexity problem for $\operatorname{GL}_2$ over any fixed number field was recently solved by Michel and Venkatesh in a beautiful preprint [@MV], where the reader can also find detailed references to previous work done in the subject. Yet, as the authors of [@MV] remark, their emphasis was not on obtaining best exponents but rather finding some nontrivial exponent that works in all cases. The aim of the present paper is to demonstrate that a relatively strong Burgess-type subconvexity bound in the conductor aspect can be achieved for the family at hand, and once the background on automorphic forms has been set up, the proof requires comparatively little effort (cf. Section \[heuristic\]).

More precisely, let $\pi$ be an irreducible cuspidal representation of $\operatorname{GL}_2( K) \backslash \operatorname{GL}_2(\mathbb{A}_K)$ with unitary central character, and let $\chi$ be a Hecke character of conductor ${\mathfrak{q}}$. Let $L(s,\pi \otimes \chi)$ denote the twisted $L$-function. Let $0{\leqslant}\theta{\leqslant}1/2$ 