High Weight

Farmer and James (see [CF99] and [FJ02]) verified that the characteristic polynomials of a very impressive range of Hecke operators $T_{p,k}$ on level one modular forms are irreducible.

Problem 1.3.1   Greatly extend the range of computation carried out by Farmer and James. For example, verify that $T_2$ acts irreducibly on $S_k(1)$ for all $k\leq 4096$ .