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PROG
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(PARI) \\ See Morison theorem 2.6
\\ F(n, q, k) is number of solutions to X^k=I in GL(i, GF(q)) for i=1..n.
\\ q is power of prime and gcd(q, k) = 1.
B(n, q, e)={sum(m=0, n\e, x^(m*e)/prod(k=0, m-1, q^(m*e)-q^(k*e)))}
F(n, q, k)={if(gcd(q, k)<>1, error("no can do")); my(D=ffgen(q)^0); my(f=factor(D*(x^k-1))); my(p=prod(i=1, #f~, (B(n, q, poldegree(f[i, 1])) + O(x*x^n))^f[i, 2])); my(r=B(n, q, 1)); vector(n, i, polcoeff(p, i)/polcoeff(r, i))}
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