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Let f be a nonzero cusp form of even integral weight k ⩾ 2 on the Hecke congruence subgroup Γ 0 ( N ) with N squarefree. Suppose that the normalized Fourier coefficients λ f ( n ) of f are real. We prove that the first sign change of λ f ( n ) occurs in the range n ≪ ( k N ) 2 + ε . This improves upon the earlier result of Choie and Kohnen [2] . [ABSTRACT FROM AUTHOR]

Let λf (n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∊ Sk(Г). In this paper, we study the asymptotics for double sum involving Fourier coefficient λi f (n) of cusp forms, where i = 2, 4, 6, 8. [ABSTRACT FROM AUTHOR]

In this paper we consider the mean value of hybrid arithmetic function of the form λ6f (n)σb(n)Øc(n), where λf (n) is the n-th Fourier coefficients of the holomorphic cusp form f, σ(n) is the sum-of-divisors function, Ø(n) is the Euler's totient function. In detail, we prove that for any e > 0, ∑ n⩽x λ6f (n)σb(n)Øc(n) = x(b+c+1)P4(log x) + O(xb+c+631/652+ε); where b; c ∊ ℝ. [ABSTRACT FROM AUTHOR]

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]