Sparse Distribution of Coefficients of $\ell$-fold Product $L$-functions at Integers Represented by Quadratic Forms

Abstract

Let $f \in S_{k}(\Gamma_{0}(N))$ be a normalized Hecke eigenform. We study the Fourier coefficients $\lambda_{f \otimes \cdots \otimes_{\ell} f}(n)$ of the $\ell$-fold product $L$-function for odd $\ell \ge 3$. Our focus is the distribution of this sequence over the sparse set of integers represented by a primitive, positive-definite binary quadratic form $Q$ of a fixed discriminant $D$. We establish an explicit upper bound for the summatory function of these coefficients, with dependencies on the weight, level, and discriminant. As a key application, we provide a bound for the first sign change of the sequence in this setting. We also generalize this result to find the first sign change among integers represented by any of the $h(D)$ forms of discriminant $D$, showing the bound improves as the class number increases.