“…In [7] and [2], the authors first show that certain of the T 1,1 2,k are irreducible and then using effective versions of the Chebotarev density theorem to show that a positive proportion of the T 1,1 l,k are irreducible, and in [1], plausible hypotheses on the non-vanishing of certain coefficients of modular forms are used to prove irreducibility of the Hecke polynomials (when the weight is 12, for instance, these hypotheses are exactly Lehmer's conjecture on the Ramanujan τ -function).…”