On some estimates involving Fourier coefficients of Maass cusp forms

Let f be a Hecke-Maass cusp form for SL2(Z) with Laplace eigenvalue lambda(f)(Delta) = 1/4+mu(2) and let lambda(f)(n) be its nth normalized Fourier coefficient. It is proved that, uniformly in alpha,beta is an element of R, Sigma(n <= X)lambda(f)>ne(alpha n(2)+beta n) << X7/8+epsilon lambda(f)(Delta)(1/2+epsilon), depends only on epsilon. We also consider the summation function of lambda(f) (n) and under the Ramanujan conjecture we are able to prove Sigma(n <= X)lambda(f)(n) << X1/3+epsilon lambda(f)(Delta)(4/9+epsilon )the implied constant depending only on epsilon.