Period polynomials and congruences for Hecke algebras

Using the Eichler-Shimura isomorphism and the action of the Hecke operator T-2 on period polynomials, we shall give a simple and new proof of the foll owing result (implicitly contained in the literature): let f be a normalize d Hecke eigenform of weight k with respect to the full modular group with e igenvalues lambda(p) under the usual Hecke operators T-p (p a prime). Let K -f be the field generated over Q by the lambda(p) for all p. Let P be a pri me of K-f lying above 5. Then lambda 2 not equal 0 (mod P). 1991 Mathematics subject classification. 11F.