Mathematical Research Letters 4, 295–307 (1997) ON THE COEFFICIENTS OF CUSP FORMS Dinakar Ramakrishnan To Robert Langlands 1. Introduction Let F be a number field, and let π be a cuspidal, unitary automorphic rep- resentation of GL(n, A F ). For almost all finite places v, let A v (π) denote the associated (Langlands) conjugacy class in GL(n, C), which is represented by a diagonal matrix [α 1,v ,...,α n,v ]. The trace of this class is denoted a v . The gen- eral Ramanujan (or purity) conjecture predicts that each α j,v has absolute value 1. This is clearly true for n =1, and a series of deep theorems asserts that it holds for (n =2,F totally real) if π corresponds to a holomorphic eigenform ([D], [DS], [BL], [C], [W], [T1], [BR]). On the other hand, one knows unconditionally that |α j,v | is bounded by (Nv) 1/2-1/(n 2 +1) for any n ([LRS]), in fact (strictly) by (Nv) 1/5 for n = 2 and any F ([Sh 1,2]). For GL(2)/Q, one has the still stronger bound |α j,v |≤ (Nv) 5/28 ([BDHI]; see also [LRS]). If the analyticity of (twists of) the symmetric fourth power L-functions of GL(2) is estabished, and some progress has been made in this direction by D. Ginzburg, then one would be able to replace 5/28 by 1/6. Striking and deep though these results are, we are still far from the conjecture. In this paper we certainly do not prove the conjecture, but we try to approach the problem from another direction. Our object is to understand better, for GL(2), the structure of the set of primes v in F where the conjecture does hold. For n ≥ 3, we still get information on the trace a v , but this is weaker than knowing the conjecture. For any set X of primes, denote by δ (X ) (resp. δ(X )) the lower (resp. upper) Dirichlet density (see below) of X, so that δ (X ) ≤ δ(X ), with equality holding iff X has a density. Our main result is Theorem A. For every cuspidal, unitary automorphic representation π of GL(n, A F ), let S (π) denote the set of primes where |a v |≤ n. If n =2, we have δ (S (π)) ≥ 9/10. If n ≥ 3, then δ (S (π)) ≥ 1 - 1/n 2 . One sees easily that, for GL(2), S (π) is precisely the set of primes where the Ramanujan conjecture holds (cf. Remark 4.10). Received October 28,1996. Supported by National Science Foundation grant DMS-9501151. 295