Uniform decay for quadratic Hecke sums

Let $(\varphi_k)_k$ be the sequence of holomorphic Hecke eigenforms under consideration, with Hecke eigenvalues $\lambda_{\varphi_k}(n)$. Let $\mathcal{S}_N$ denote the Sobolev norms on $C_c^\infty(\mathbb{R}_{+}^{\times})$, and let $\|Q_k\|$ denote the coefficient norm of an irreducible integer-valued quadratic polynomial $Q_k$. Assume that the $\varphi_k$ are non-dihedral. Quadratic Hecke-sum conjecture. There exists $A\geqslant 0$ such that, for every sequence of irreducible integer-valued quadratic polynomials $Q_k$ and test functions $f_k\in C^\infty(\mathbb{R}_{+}^{\times})$ satisfying $\mathcal{S}_N(f_k)\ll_N1$ for each fixed $N$,

$$\lim_{k\to\infty}\frac{\sum_n\lambda_{\varphi_k}(|Q_k(n)|)f_k(n/k)}{kL(\operatorname{ad}\varphi_k,1)\|Q_k\|^A}=0.$$

This conjecture extrapolates the available bounds for quadratic Hecke sums to sequences in which both the eigenform and summation length vary. For full-level forms, the non-dihedrality assumption is automatic; the source notes that the higher-level generalization requires it explicitly.