Deterministic comparison of cusp form coefficients over certain sequences

Let f and g be two distinct primitive holomorphic cusp forms of even integral weights \(k_{1}\) $k_1$ and \(k_{2}\) $k_2$ for the full modular group \(\Gamma =SL(2,{\mathbb {Z}})\) $\mathrm{\Gamma }=SL(2,\mathbb{Z})$ , respectively. Denote by \(\lambda _{f}(n)\) ${\lambda }_f(n)$ and \(\lambda _{g}(n)\) ${\lambda }_g(n)$ the nth normalized Fourier coefficients of f and g, respectively. And set \(Q(\textbf{x})\) $Q(\mathbf{x})$ a primitive integral positive-definite binary quadratic form of fixed discriminant \(D<0\) $D<0$ with the class number \(h(D)=1\) $h(D)=1$ . In this paper, we establish a lower bound for the analytic density of the set \(\begin{aligned} \big \{ p ~: ~ p=Q(\textbf{x}) ~ \text { for some }~ \textbf{x}\in {\mathbb {Z}}^{2}, ~\lambda _{f}(p^{i})\lambda _{f}(p^{j}) < \lambda _{g}(p^{i})\lambda _{g}(p^{j})\big \}, \end{aligned}\) $$\begin{array}{c}\hfill \{p:p=Q(\mathbf{x})\text{for some}\mathbf{x}\in {\mathbb{Z}}^2,{\lambda }_f(p^i){\lambda }_f(p^j)<{\lambda }_g(p^i){\lambda }_g(p^j)\},\end{array}$$ where \(j\geqslant 1, 0\leqslant i\leqslant j\) $j⩾1,0⩽i⩽j$ are any fixed integers. Furthermore, we also consider the similar problem concerning the linear combinations of \(\lambda _{f}(p^{j})\) ${\lambda }_f(p^j)$ and \(\lambda _{g}(p^{j})\) ${\lambda }_g(p^j)$ in a given interval supported on the binary quadratic form \(Q(\textbf{x})\) $Q(\mathbf{x})$ . Similar problems for triple product L-functions are also studied.