Level $1$ Weight $24$

Gather data and theoretical results about the following question, which Ralph Greenberg asked (though surely other people have asked it):

Problem 1.1.1   Is the characteristic polynomial of every Hecke operator $T_p$ irreducible on the 2-dimensional space of cusp forms of level 1 and weight 24? (Stein checked that it is for all primes up to 800 and for p=144169.)

The rest of this section was written by Koopa Koo, and explains some results related to (1.1.1).

The goal of this section is to explain how to apply Chebotarev's density theorem to obtain density result about irreducibility of the characteristic polynomial of the Hecke operators attached to the weight 24 cusp forms of level 1, $S_{24}(\Gamma_0(1)).$

I would like to thank my advisor Prof. Ralph Greenberg for suggesting the problem with helpful suggestions, Prof. Gabor Wiese for his helpful suggestions, and my greatest thanks to Prof. William Stein, who advised the project and provided many inspiring ideas.

Problem 1.1.2   I expect the same method generalizes to higher weight and higher level, by working with mod $\lambda$ representations. Help me carry out this generalization and write up the details nicely.