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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4080, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([4,4,4,4,1]))
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pari:[g,chi] = znchar(Mod(3239,4080))
\(\chi_{4080}(359,\cdot)\)
\(\chi_{4080}(1079,\cdot)\)
\(\chi_{4080}(3239,\cdot)\)
\(\chi_{4080}(3959,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((511,3061,1361,817,241)\) → \((-1,-1,-1,-1,e\left(\frac{1}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4080 }(3239, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
