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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9576, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,0,3,15,8]))
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pari:[g,chi] = znchar(Mod(47,9576))
\(\chi_{9576}(47,\cdot)\)
\(\chi_{9576}(815,\cdot)\)
\(\chi_{9576}(4583,\cdot)\)
\(\chi_{9576}(6599,\cdot)\)
\(\chi_{9576}(7871,\cdot)\)
\(\chi_{9576}(9383,\cdot)\)
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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 ]
\((7183,4789,5321,4105,1009)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{5}{6}\right),e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 9576 }(47, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) |
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sage:chi.jacobi_sum(n)
