Let f be a Maass cusp form for Γ0(N) with Fourier coefficients λf(n) and Laplace eigenvalue \(\frac{1} {4} + k^2 \). For real α ≠ 0 and β > 0, consider the sum SX(f; α, β) = ∑n λf(n)e(αnβ)ϕ(n/X), where ϕ is a smooth function of compact support. We prove bounds for the second spectral moment of SX(f; α, β), with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X. This implies that if f has its eigenvalue beyond \(X^{\tfrac{1} {2} + \varepsilon } \), the standard resonance main term for SX(f; \( \pm 2\sqrt q \), 1/2), q ∈ ℤ+, cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL(2) × GL(2). It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of Ke ⩽ L ⩽ K1−e. The same bounds can be proved in a similar way for holomorphic cusp forms.