Resonance between automorphic forms and exponential functions

Let f be a holomorphic cusp form of weight k for SL ₂(?) and λ _f (n) its n-th Fourier coefficient. In this paper, the exponential sum Σ _{X < n ? 2X} λ _f (n)e(αn ^B ) twisted by Fourier coefficients λ _f (n) is proved to have a main term of size |λ _f (q)|X ^3/4 when β = 1/2 and α is close to $ \pm 2\sqrt q ,q \in \mathbb{Z} $ , and is smaller otherwise for β < 3/4. This is a manifestation of the resonance spectrum of automorphic forms for SL ₂(?).