Summary: Let \(f\) be either a holomorphic Hecke eigenform of weight \(\kappa \) for \(\text{SL}_2(\mathbb Z)\) with \[ f(z) = \sum ^{\infty }_{n=1}\lambda (n)n^{(\kappa -1)/2}e(nz), \] or a Maass Hecke eigenform for \(\text{SL}_2(\mathbb Z)\) with Laplace eigenvalue \(\frac{1}{4} + \nu ^{2}\). In the latter case, \[ f(z) = 2\sqrt{y} \sum _{n \neq 0} \rho (n)K_{i\nu }(2\pi |n|y)e(nx). \] Here \(K_{i\nu }\) is the modified Bessel function of the third kind and \(e(z) = e^{2\pi iz}\). This paper studies the cancellation of the coefficients \(\lambda (n)\) or \(\rho (n)\) in nonlinear exponential sums with amplitude \(n^{\theta }, 0 < \theta \leq \frac{1}{2}\).