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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1120, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,7,4,0]))
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pari:[g,chi] = znchar(Mod(589,1120))
\(\chi_{1120}(29,\cdot)\)
\(\chi_{1120}(309,\cdot)\)
\(\chi_{1120}(589,\cdot)\)
\(\chi_{1120}(869,\cdot)\)
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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 ]
\((351,421,897,801)\) → \((1,e\left(\frac{7}{8}\right),-1,1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1120 }(589, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
