## Abstract

Let f(z) = ∑^{∞}_{n=1} a(n)e^{2πinz} ∈ S^{new}k_{} (Γ(N)) be a newform of even weight k ≥ 2 that does not have complex multiplication. Then a(n) ∈ ℝ for all n; so for any prime p, there exists θ_{p} ∈ [0, π] such that a(p) = 2p^{(k-1)/2} cos(θ_{p}).Let π(x) = ≠ {p ≤ x}. For a given subinterval [α, β] ⊂ [0, π], the now-proven Sato-Tate conjecture tells us that as x → ∞. # {p ≤ x: θ_{p} ∈ I} ~ μST([α, β]) π (x), μST([α, β]) = ∫^{β}_{α} 2\πsin^{2}(θ) dθ.Let ∈ > 0. Assuming that the symmetric power L-functions of f are automorphic, we prove that as x → ∞, #{p ≤ x: θ_{p} ∈ I} = μST([α, β])π(x) + O (x/(log x)^{9/8- ∈}) where the implied constant is effectively computable and depends only on k,N, and ∈.

Original language | English (US) |
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Pages (from-to) | 147-156 |

Number of pages | 10 |

Journal | Archiv der Mathematik |

Volume | 103 |

Issue number | 2 |

DOIs | |

State | Published - Aug 2014 |

Externally published | Yes |

## Keywords

- Automorphic forms
- Sato-Tate conjecture

## ASJC Scopus subject areas

- General Mathematics