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

Let $sy{m}^{2}f$ denote the symmetric square lift of a Hecke eigenform $f\in S_k({\mathrm{\Gamma }}_0(N))$ with the $n^{\mathrm{t}\mathrm{h}}$-Fourier coefficients ${\lambda }_{sy{m}^{2}f}(n)$. In this article, we prove an estimate for the first moment of the sequence $\{{\lambda }_{sy{m}^{2}f}(\mathcal{Q}(\underset{‾}{x})){\}}_{\mathcal{Q}\in {\mathcal{S}}_D,\underset{‾}{x}\in {\mathbb{Z}}^2}$ 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 }} λ{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 ${\lambda }_{sy{m}^{2}f}(n),$ supported on the set of integers represented by reduced forms of the discriminant $D$. We determine the size of $n_{sy{m}^{2}f,D}$ (see definition before \thmref{ExtMatKLSW}), in terms of the conductor of the associated $L$-functions.