For each prime $p$, we determine the distribution of the $p^{th}$ Fourier
coefficients of the Hecke eigenforms of large weight for the full modular
group. As $p\to\infty$, this distribution tends to the Sato--Tate distribution

Conrey, J. Brian

Duke, William

Farmer, David W.

English

arXiv

. For each prime p, we determine the distribution of the p  th  Fourier coefficients of the Hecke eigenforms of large weight for the full modular group. As p !1,  this distribution tends to the Sato--Tate distribution. 1. Introduction and statement of results  Let  f(z) =  1  X  n=1  a f;n e(nz) be a cusp form of weight k for the full modular group. We assume that f(z) is an eigenfunction of all the Hecke operators T n , and we set a f (n) = n  1\Gammak 2  a f;n . The assumption that f(z) is a Hecke eigenform gives:  a f (mn) = a f (n)a f (m) if (m; n) = 1 a f (p  j  ) = a f (p)a f (p  j \Gamma1  ) \Gamma a f (p  j \Gamma2  ) for p prime ja f (n)j  d(n); where d(n) is the divisor function. The above relations were first conjectured by Ramanujan based on a computation of the coefficients of the weight 12 cusp form \Delta(z) =  P  ø (n)e(nz). The two equations were proven for ø (n) by Mordell, using what are now known as the Hecke operators. The inequality was proven by Deligne as a cons..

J. B. Conrey

W. Duke

D. W. Farmer

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The Distribution of the Eigenvalues of Hecke Operators

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.9266

The distribution of the eigenvalues of Hecke operators

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