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

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.