Twisted Böcherer conjecture for squarefree-level 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 $D$ with $\ell D<0$, let $B_{\ell,F}(D)$ be the twisted average of Fourier coefficients indexed by quadratic forms of discriminant $D$. If $\Delta_0$ is the fundamental discriminant associated to $\Delta$, define

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

Twisted Böcherer conjecture.

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

where $C_{\ell,F}$ is independent of $D$, and $C_{\ell,F}=0$ if and only if $L(F,1/2,\ell)=0$. This generalizes the prime-level paramodular form of Böcherer's conjecture by allowing an auxiliary discriminant and arbitrary squarefree level; the notation and conjectural relation are presented as the basis for numerical tests on nonlifts.