Eigenvalue-weighted moment conjecture for holomorphic cusp forms

Let $\ell\geq1$ and $r\geq1$ be integers, and set

$$X_k(s)=\frac{\Gamma\left(\frac{1}{2}-s+\frac{k}{2}\right)}{\Gamma\left(s-\frac{1}{2}+\frac{k}{2}\right)}.$$

For the forms $f\in H_k(1)$, let $\lambda_f(\ell)$ denote the Hecke eigenvalue, let $L(s,f)$ be the associated $L$-function, and let $L(s,\operatorname{sym}^2 f)$ be the symmetric-square $L$-function. Define

$$G(\ell;z_1,\ldots,z_r)=A(\ell;z_1,\ldots,z_r)\prod_{1\leq i<j\leq r}\zeta(1+z_i+z_j),$$

where $A$ is the absolutely convergent arithmetic factor defined in the source, and let $\Delta$ denote the Vandermonde determinant. Eigenvalue-weighted moment conjecture. There is a positive constant $\delta$ such that

$$\sum_{f\in H_k(1)}\frac{2\pi^2\lambda_f(\ell)L\left(\frac{1}{2},f\right)^r}{(k-1)L(1,\operatorname{sym}^2 f)}=R_{k,r}(\ell)\left(1+O\left((\ell k)^{-\delta+\varepsilon}\right)\right),$$

where

$$R_{k,r}(\ell)=\frac{(-1)^{r(r-1)/2}2^r}{r!}\frac{1}{(2\pi i)^r}\oint\cdots\oint\frac{G(\ell;z_1,\ldots,z_r)\Delta(z_1^2,\ldots,z_r^2)^2}{z_1^{2r-1}\cdots z_r^{2r-1}}\prod_{j=1}^rX_k\left(\frac{1}{2}+z_j\right)^{-1/2}\,dz_1\cdots dz_r.$$

The formula is a conjectural asymptotic for eigenvalue-weighted central moments, motivated by the recipe of Conrey, Farmer, Keating, Rubinstein and Snaith; its validity in the stated generality remains open.