ON CUSP FORM COEFFICIENTS IN NONLINEAR EXPONENTIAL SUMS

Let f be either a holomorphic Hecke eigenform of weight κ for SL2(ℤ) with $\cal f (z) = \sum_{n=a}^\infty \lambda (n)n^{k-a)/2}e(nz),$ or a Maass Hecke eigenform for SL2(ℤ) with Laplace eigenvalue ¼ + ν². In the latter case, ${\cal f} (z) = 2\sqrt{y}\sum_{n\neq0}\rho(n)k_{iv}(2\pi\mid n\mid y) e(nx).$ Here Kiν is the modified Bessel function of the third kind and e(z) = e^2πiz. This paper studied the cancelation of the coefficients λ(n) or ρ(n) in nonlinear exponential sums with amplitude n^θ, 0 < θ≤ ½.