Fourier coefficients of forms of CM-type

Let f be a cuspidal normalized eigenform of weight ≥ 2 for Г 0 ( N ),with Fourier expansion While the Galois representations associated to f can be used effectively to study the divisibility properties of the Fourier coefficients, it is very difficult to analyze the condition a f ( p ) = 0 (mod p ). In this paper, we show that the problem is accessible in the case that f has complex multiplication. Under some mild conditions on f , we show that for p sufficiently large, a f ( p ) = 0 (mod p ) in fact implies that a f ( p ) = 0.