Oscillations of Fourier coefficients over the sparse set of integers

Let \(f \in S_{k}(\Gamma _{0}(N))\) $f\in S_k({\mathrm{\Gamma }}_0(N))$ be a normalized Hecke eigenforms of integral weight k and level \(N \ge 1\) $N\ge 1$ . In the article, we establish the asymptotics of power moment associated to the sequences \(\{\lambda _{f \otimes f \otimes f}(\mathcal {Q}(\underline{x}))\}_{\mathcal {Q} \in \mathcal {S}_{D}, \underline{x} \in \mathbb {Z}^{2}}\) ${\{{\lambda }_{f\otimes f\otimes f}(\mathcal{Q}(\underline{x}))\}}_{\mathcal{Q}\in {\mathcal{S}}_D,\underline{x}\in {\mathbb{Z}}^2}$ and \(\{\lambda _{f \otimes \mathrm{sym^{2}}f}(\mathcal {Q}(\underline{x}))\}_{\mathcal {Q} \in \mathcal {S}_{D}, \underline{x} \in \mathbb {Z}^{2}}\) ${\{{\lambda }_{f\otimes {\mathrm{sym}}^{2}f}(\mathcal{Q}(\underline{x}))\}}_{\mathcal{Q}\in {\mathcal{S}}_D,\underline{x}\in {\mathbb{Z}}^2}$ where \(\mathcal {S}_{D}\) ${\mathcal{S}}_D$ denotes the set of inequivalent primitive integral positive-definite binary quadratic forms (reduced forms) of fixed discriminant \(D < 0.\) $D<0.$ As a consequence, we prove results concerning the behaviour of sign changes associated to these sequences.