On some exponential sums involving Maass forms over arithmetic progressions

Let g(z)g(z) be a Maass cusp form for SL(2,Z)SL(2,Z), and let λg(n)λg(n) be the n  -th Fourier coefficient of g(z)g(z). In this paper we investigate the nonlinear exponential sum∑n∼Xn≡lmodqλg(n)e(αnβ) twisted by Fourier coefficient over the arithmetic progression. We prove an asymptotic formula when β=1/2β=1/2 and α   is close to ±2kq,k∈Z+.