On the simultaneous sign changes of Hecke eigenvalues associated with multiple cusp forms

Let f, g and h be three distinct normalized primitive Hecke cusp forms of even integral weights \(k_{1},k_{2}\) $k_1,k_2$ and \(k_{3}\) $k_3$ for the full modular group \(\Gamma =SL(2,{\mathbb {Z}})\) $\mathrm{\Gamma }=SL(2,\mathbb{Z})$ , respectively. Denote by \(\lambda _{f}(n),\lambda _{g}(n)\) ${\lambda }_f(n),{\lambda }_g(n)$ and \(\lambda _{h}(n)\) ${\lambda }_h(n)$ the n-th normalized Fourier coefficients of f, g and h, respectively. Let \(Q({\varvec{x}})\) $Q(\mathit{x})$ denote a certain primitive integral binary quadratic form with negative discriminant \(D<0\) $D<0$ . In this paper, we consider the number of sign changes of the sequence \(\{\lambda _{f}(n)\lambda _{g}(n)\lambda _{h}(n)\}_{n\geqslant 1}\) ${\{{\lambda }_f(n){\lambda }_g(n){\lambda }_h(n)\}}_{n⩾1}$ supported at \(Q({\varvec{x}})\) $Q(\mathit{x})$ in the interval (x, 2x], where \(x>0\) $x>0$ is a sufficiently large real number.