Introduction. Let N 2 1, k 2 2 be integers and let f be a cusp form of weight k for ro(N), which is a normalized eigenform for all the Hecke operators Tp, p 4 N. Let us write f (z) == an 2winz n>l for its Fourier expansion at ioo, and for simplicity of discussion, let us suppose that the an are rational integers. For each integral value of a, set 7rf,a(X) = #{P s x :aap = a}. If a = 0 and f is of CM-type (in the sense of Ribet [11]), we know that lrf,a(X) r(x)/2. In the remaining cases, (i.e. f is not of CM type or a * 0), Lang and Trotter [5] conjecture that l/2/log x if k = 2 Wf,a(X)-Cf,a loglog x if k = 3 1 if k 2 4 where Cf a is a constant which is generally (though not always) nonzero (Cf,a can be zero for example if a = 1, k = 2 and f corresponds to an elliptic curve having a nontrivial Q-rational torsion point). Moreover, Atkin and Serre [15] conjecture that for k 2 4, Manuscript received 4 April 1986; revised 6 April 1987. '.2Research partially supported by NSERC grants.