Explicit Burgess-like subconvex bounds for ${\mathrm{G}\mathrm{L}}_2\times {\mathrm{G}\mathrm{L}}_1$

We make the polynomial dependence on the fixed representation $\pi$ in our previous subconvex bound of $L(1\mathrm{/}2,\pi \otimes \chi )$ for $$\mathrm{G}\mathrm{L}$_2 \times $\mathrm{G}\mathrm{L}$1$explicit,especiallywithrespecttotheusualconductor$$\mathbf{C}$(\pi{$\mathrm{f}\mathrm{i}\mathrm{n}$})$. There is no clue that the original choice, due to Michel & Venkatesh, of the test function at the infinite places should be the optimal one. Hence we also investigate a possible variant of such local choices in some special situations.