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Fast characteristic polynomial algorithm over cyclotomic fields

Using this code with 23 replaced by 23, 67, and 199 I benchmarked both Sage and Magma on various machines. Here Sage is just using PARI. For 23, Magma is 10 times faster. For 67, Magma is 5-10 times faster. For 199, Magma and Sage are almost the same. So probably they are using exactly the same algorithm, but Magma is more optimized for the small cases.

ModularSymbols_clear_cache()
eps = DirichletGroup(23*3, CyclotomicField(11)).1^2
M = ModularSymbols(eps); M
t = M.hecke_matrix(2)

The timings themselves are as follows, all on an Opteron 1.8Ghz.

Similar timings for computing the square of the matrix T_2, i.e., for matrix multiplication:

So Magma's matrix multiply is vastly superior to what is in Sage, but only because Sage's multiply is very stupid and generic.