Question: Most efficient way to get eigenvalues

Hi all. I am trying to use the Weyl character formula to evaluate some traces of Hecke operators on algebraic modular forms. This involves me having to substitute certain eigenvalues of 4x4 matrices into a polynomial of arbitrary degree (my choice).

I have created procedures that do this given the eigenvalues. Unfortunately I am having problems generating the eigenvalues.

The problem is that the matrices I am using are formed from products of matrices from two sets, one containing 40 matrices and one containing 1152 matrices. So in total I am having to get the Eigenvalues of 46080 matrices.

I have tried for a while to generate these efficiently but I am not succeeding (usually leave it running for hours and get nowhere which should not be the case should it?). What is the most efficient way I can do this? Should I generate the matrices first in lumps or all in one and then do eigenvalues or find the eigenvalues with each product straight away?

I have been experimenting with JordanForm instead of Eigenvalues (this command seems to work faster but requires more storage). I also might be able to work with decimal approximation since I expect the result of my calculation to be an integer.