Parametric twisted Böcherer conjecture for paramodular forms

Let $N$ be squarefree, and let $F\in S^k(\Gamma^{\mathrm{para}}[N])$ be a Hecke eigenform that is not a Gritsenko lift. For fundamental discriminants $\ell$ and $D$ satisfying $\ell D<0$, let $B_{\ell,F}(D)$ be the corresponding twisted Fourier-coefficient average, and define $\alpha_{\ell D}$ by

$$\alpha_\Delta=\prod_{p\mid N}\left(1+\left(\frac{\Delta_0}{p}\right)\right),$$

where $\Delta_0$ is the fundamental discriminant associated to $\Delta$. Parametric twisted Böcherer conjecture. There is a positive constant $k_F$, independent of $\ell$ and $D$, such that

$$B_{\ell,F}(D)^2=\alpha_{\ell D}k_F L(F,1/2,\chi_\ell)L(F,1/2,\chi_D)|D\ell|^{k-1}.$$

This conjecture refines the preceding twisted formulation by making the dependence on the auxiliary discriminant explicit. It is proposed for nonlift eigenforms and is supported in the paper by the numerical examples discussed there.