  Let T^{N,chi}_{p,k}(x) be the characteristic polynomial of the Hecke operator
T_p acting on the space of cusp forms S_k(N,chi). We describe the factorization
of T^{N,chi}_{p,k}(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^{1,1}_{q,k}(x) from the irreducibility of
T^{1,1}_{2,k}(x).
