Average behavior of Fourier coefficients of Maass cusp forms for hyperbolic -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) Sigma(N(n)<= x)lambda(phi)(n)(j), where j <= 8; (ii) the sum of lambda(phi)(n) over the sparse sequence n(l) Sigma(N(n)<= x)lambda(phi)(n)(l), where l <= 4; (iii) the hybrid sum for lambda(phi)(n) Sigma(N(n)<= x)lambda(phi)(n(l))(j), where 2 <= l <= 4, j = 2, or l = 2, j = 4.