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How to compute the eigenvalue of a matrix over a finite field?

I would like to compute the eigenvalues of the following matrix over $\mathbb F_5$.

Eigenvalues[{{2,-2,0,-1,-1},{2,2,-2,2,-2},{0,0,-1,1,0},{0,0,1,2,-2},{0,0,-2,2,2}}, Modulus->5]

But I get the error output:

Eigenvalues :UnknownoptionModulusinEigenvalues[{{2,-2,0,-1,-1},{2,2,-2,2,-2},{0,0,-1,1,0},{0,0,1,2,-2},{0,0,-2,2,2}},Modulus5].

Question: Is calculating the eigenvalue over a finite field possible in mathematica?

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    $\begingroup$ Take a look at a closely related issue Finding the characteristic polynomial of a matrix modulus n $\endgroup$ – Artes Jul 4 '17 at 19:03
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    $\begingroup$ In addition to the remark by @Artes, also note the roots of the char poly do not all lie in the base field: In[179]:= Factor[ CharacteristicPolynomial[{{2, -2, 0, -1, -1}, {2, 2, -2, 2, -2}, {0, 0, -1, 1, 0}, {0, 0, 1, 2, -2}, {0, 0, -2, 2, 2}}, x], Modulus -> 5] Out[179]= 4 (2 + x)^2 (4 + x) (3 + x^2) $\endgroup$ – Daniel Lichtblau Jul 4 '17 at 20:01

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