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Let f be a Hecke eigencusp form of even integral weight k or Maass cusp form for the full modular group S L 2 ( Z ) . Denote by λ sym m f ( n ) the n th normalized coefficient of the Dirichlet expansion of the m th symmetric power L -function associated to f . In this paper, we establish some bounds for ∑ n ⩽ x λ sym m f ( n ) , ∑ n ⩽ x λ f ( n m ) , which improve the corresponding results of Lau & Lü [8] . [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]

Let f be a primitive holomorphic or Maass cusp form for the group SL ( 2 , Z ) , and a f ( n ) its n th normalized Fourier coefficient. It is proved that, for any α , β ∈ R , there exists an effective positive constant c such that ∑ n ≤ N Λ ( n ) a f ( n ) e ( α n 2 + β n ) ≪ f N exp ⁡ ( − c log ⁡ N ) . [ABSTRACT FROM AUTHOR]