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Let g be a holomorphic or Maass Hecke newform of level D and nebentypus χD, and let a ( n) be its n-th Fourier coefficient. We consider the sum $${S_1} = \sum {_{X < n \leqslant 2X}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)}$$ and prove that S has an asymptotic formula when β = 1/2 and α is close to $$\pm 2\sqrt {q/D}$$ for positive integer q ≤ X/4 and X sufficiently large. And when 0 < β < 1 and α, β fail to meet the above condition, we obtain upper bounds of S . We also consider the sum $${S_2} = \sum {_{n > 0}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)\phi \left( {n/X} \right)}$$ with ø( x) ∈ C (0,+∞) and prove that S has better upper bounds than S at some special α and β. [ABSTRACT FROM AUTHOR]

In this paper, we study the exponential sums involving Fourier coefficients of cusp forms for SL ( 2 , Z ) over primes. We are able to establish some new results. In addition, similar results have been obtained for the coefficients of symmetric-square lifts. [ABSTRACT FROM AUTHOR]