Let $S_k(N,\chi)$ denote the space of holomorphic cusp forms of integral weight $k$ and character $\chi$ for $\Gamma_0(N)$ and let $T^{N,\chi}_{p,k}(x)$ denote the characteristic polynomial of the Hecke operator $T_p$ acting on $S_k(N,\chi)$. \par Maeda has conjectured that the Hecke algebra of $S_{2k}(1)$ over $\Bbb Q$ is simple, and that its Galois closure over $\Bbb Q$ has Galois group $\germ S_{d_k}$. Recently there have been numerous investigations regarding the irreducibility of characteristic polynomials of Hecke operators on $S_{2k}(1)$. The existence of such polynomials has proven to be useful in proving nonvanishing theorems for central values of level 1 modular $L$-functions, and in constructing base changes to totally real number fields for level 1 eigenforms [see, e.g., J. B. Conrey\ and D. W. Farmer, in {\it Topics in number theory (University Park, PA, 1997)}, 143--150, Kluwer Acad. Publ., Dordrecht, 1999; MR1691315 (2000f:11055); H. Hida\ and Y. Maeda, Pacific J. Math. {\bf 1997}, Special Issue, 189--217; MR1610859 (99f:11068); W. Kohnen and D. Zagier, Invent. Math. {\bf 64} (1981), no.~2, 175--198; MR0629468 (83b:10029)]. \par In the paper under review the authors describe a finite calculation which determines the factorization of $T^{N,\chi}_{n,k}(x)$ modulo a prime $ l$ for any $k$. They employ this method to prove that if $T^{1,1}_{n,k}(x)$ is irreducible and has full Galois group for some $n$, then $T^{1,1}_{p,k}(x)$ is irreducible and has full Galois group for $p$ any prime satisfying $p\not\equiv \pm 1 \pmod{5}$ and $p\not\equiv \pm 1 \pmod{7}$. (For a similar theorem for higher level, see [K. L. James\ and K. Ono, J. Number Theory {\bf 73} (1998), no.~2, 527--532; MR1658012 (2000a:11063)].) \par The reviewer and the second author have used a version of the above result to show that $T^{1,1}_{p,k}(x)$ is irreducible and has full Galois group if $p<2000$ and $k\le 2000$.