On Fourier coefficients associated to automorphic L-functions over a binary quadratic form and its applications

Let f and g be two distinct normalized primitive Hecke cusp forms of even integral weights \(k_{1}\) and \(k_{2}\) for the full modular group \(\Gamma =SL(2,{\mathbb {Z}})\), respectively. Denote by \(\lambda _{f\otimes f\otimes f\otimes g}(n)\) and \(\lambda _{\text {sym}^{2}f\otimes f\otimes g}(n)\) the nth normalized coefficients of the automorphic L-functions \(L(f\otimes f\otimes f\otimes g,s)\) and \(L(\text {sym}^{2}f\otimes f\otimes g,s)\), respectively. In this paper, we are interested in the average behavior of the coefficients \(\lambda _{f\otimes f\otimes f\otimes g}(n)\) and \(\lambda _{\text {sym}^{2}f\otimes f\otimes g}(n)\) on a primitive integral binary quadratic form with negative discriminant whose class number is 1, and we also provide the asymptotic formulae of these summatory functions. As an application, we also consider the number of sign changes of the sequences \(\{\lambda _{f\otimes f\otimes f\otimes g}(n)\}_{n\geqslant 1}\) and \(\{\lambda _{\text {sym}^{2}f\otimes f\otimes g}(n)\}_{n\geqslant 1}\) on the same binary quadratic form in short intervals.