the subconvex exponent is critical, an example being [@CCU] where $L(1/2, \pi \otimes \chi) \ll (\text{cond }
\chi)^{\frac{1}{2} - \frac{1}{16}+{\varepsilon}}$ or an equivalent bound for metaplectic Fourier coefficients (over ${\mathbb{Q}}$) is needed.

An inspection of the proof shows that with somewhat more precise estimates the implied constant in Theorem \[theorem1\] turns out to be polynomial in the analytic conductor of $\pi$ and the archimedean parameters of $\chi$ with an exponent depending on ${\varepsilon}$.

The proof of Theorem \[theorem1\] builds on the ideas of several earlier works, most notably of [@DFI1; @CPSS; @V1; @Ve; @BH]. Applying an approximate functional equation, a typical off-diagonal term in the amplified second moment is essentially of the form with a slightly more general summation condition $\ell_1 n - \ell_2
m = q$ for any nonzero $q \in {\mathfrak{q}}$. Often the estimation of such expressions rests on some variant of the circle method (see e.g.[@DFI1; @BHM]) in order to detect the summation condition. However, this seems difficult to implement over number fields with a nontrivial unit and class group, in contrast to the more structural approach in [@BH] which we will follow here. The proof is written in an interesting mixture of classical and modern language: 