Average behavior of Fourier coefficients of Maass cusp forms for hyperbolic $$3$$ 3 -manifolds

Let \(\lambda _\phi (n)\) be the \(n\)-th Fourier coefficient of a doubly even and normalized Hecke–Maass cusp form for hyperbolic \(3\)-manifolds. In this paper, we investigate the behavior of summatory functions in the following (i) the \(j\)-th power sum of \(\lambda _\phi (n)\) $$\begin{aligned} \sum _{N(n)\le x}\lambda _\phi (n)^{j}, \end{aligned}$$ where \(j\le 8;\) (ii) the sum of \(\lambda _\phi (n)\) over the sparse sequence \({n^l}\) $$\begin{aligned} \sum _{N(n)\le x}\lambda _\phi (n^l), \end{aligned}$$ where \(l\le 4;\) (iii) the hybrid sum for \(\lambda _\phi (n)\) $$\begin{aligned} \sum _{N(n)\le x}\lambda _\phi (n^l)^{j}, \end{aligned}$$ where \(2\le l\le 4, j=2,\) or \(l=2, j=4.\)