Abstract

Let T_p,k^N,χ(x) be the characteristic polynomial of the Hecke operator T_p acting on the space of cusp forms S_k(N,χ). We describe the factorization of T_p,k^N,χ(x) mod ℓ as k varies, and we explicitly calculate those factorizations for N = 1 and small ℓ. These factorizations are used to deduce the irreducibility of certain T_q,k^1,1(x) from the irreducibility of T_2,k^1,1(x).

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