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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8820, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,4,9,10]))
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pari:[g,chi] = znchar(Mod(3253,8820))
\(\chi_{8820}(3253,\cdot)\)
\(\chi_{8820}(5017,\cdot)\)
\(\chi_{8820}(6793,\cdot)\)
\(\chi_{8820}(8557,\cdot)\)
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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 ]
\((4411,7841,7057,1081)\) → \((1,e\left(\frac{1}{3}\right),-i,e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8820 }(3253, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) |
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sage:chi.jacobi_sum(n)
