Oscillations of Fourier coefficients over the sparse set of integers

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