Oscillations and first-ever negative Fourier coefficients of symmetric square L-functions over sparse set

TL;DR

This paper investigates the oscillatory behavior and the occurrence of negative Fourier coefficients of symmetric square L-functions over sparse sets, providing estimates for their moments and sign patterns.

Contribution

It offers the first estimate for the first moment of Fourier coefficients over reduced forms of a discriminant and analyzes the sign distribution of these coefficients.

Findings

Established an estimate for the sum of Fourier coefficients over reduced forms.

Analyzed the sign distribution of Fourier coefficients on integers represented by quadratic forms.

Determined the size of the minimal index where negativity occurs, related to the L-function's conductor.

Abstract

Let $sym^{2} f$ denote the symmetric square lift of a Hecke eigenform $f \in S_{k}(\Gamma_{0}(N))$ with the $n^{\rm th}$-Fourier coefficients $ \lambda_{sym^{2}f}(n)$. In this article, we prove an estimate for the first moment of the sequence $\{ \lambda_{sym^{2}f}(\mathcal{Q}(\underline{x}))\}_{\mathcal{Q} \in \mathcal{S}_{D}, \underline{x} \in \mathbb{Z}^{2}}$ where $\mathcal{S}_{D}$ denotes the set of in-equivalent reduced forms of the discriminant $D$. More precisely, we establish an estimate for the following sum: \begin{equation*} \begin{split} S(sym^{2}f, D; X ) &= \sideset{}{^{\flat }}\sum_{\substack{\mathcal{Q}(\underline{x}) \leq X \\ \underline{x} \in \mathbb{Z}^{2} ,~ \mathcal{Q} \in \mathcal{S}_{D} \\ \gcd(\mathcal{Q}(\underline{x}),N) =1 }} \lambda_{sym^{2}f}(\mathcal{Q}(\underline{x})), \end{split} \end{equation*} Moreover, we consider a question concerning the behavior of signs of the Fourier coefficients $\lambda_{sym^{2}f}(n),$ supported on the set of integers represented by reduced forms of the discriminant $D$. We determine the size of $n_{sym^{2}f, D}$ (see definition before \thmref{ExtMatKLSW}), in terms of the conductor of the associated $L$-functions.