On the number of Fourier coefficients that determine a Hilbert modular form

On the Fourier coefficients of nonholomorphic Hilbert modular forms of half-integral weight

Let f(z) = ∑ n≥1 ane 2πinz be a Hecke eigenform of half-integral weightm+1/2, and let g(z) = ∑ n≥1 bne 2πinz be the corresponding even-weight form, in the sense of [Sh 73]. In particular, g has weight 2m, and belongs to the same eigenvalues of Hecke operators as f . If n = qr with squarefree r, then an is expressible in terms of ar and the {bj}. At the end of [Sh 77], Shimura suggested that ar ...

Fourier Coefficients of Modular Forms

These notes describe some conjectures and results related to the distribution of Fourier coefficients of modular forms. This is a rough draft and these notes should forever be considered incomplete.

On the Computation of the Coefficients of a Modular Form

Oscillations of Fourier Coefficients of Modular Forms

a(p) = 2p~ @ ) cos 0(p). Since we know the truth of the Ramanujan-Petersson conjecture, it follows that the 0(p)'s are real. Inspired by the Sato-Tate conjecture for elliptic curves, Serre [14] conjectured that the 0(p)'s are uniformly distributed in the interval [0, rc] with respect to the 1 measure -sin2OdO. Following Serre, we shall refer to this as the Sato-Tate r~ conjecture, there being n...

Divisors of Fourier coefficients of modular forms

Let d(n) denote the number of divisors of n. In this paper, we study the average value of d(a(p)), where p is a prime and a(p) is the p-th Fourier coefficient of a normalized Hecke eigenform of weight k ≥ 2 for Γ0(N) having rational integer Fourier coefficients.