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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(290, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([21,22]))
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pari:[g,chi] = znchar(Mod(63,290))
\(\chi_{290}(13,\cdot)\)
\(\chi_{290}(33,\cdot)\)
\(\chi_{290}(63,\cdot)\)
\(\chi_{290}(67,\cdot)\)
\(\chi_{290}(93,\cdot)\)
\(\chi_{290}(167,\cdot)\)
\(\chi_{290}(183,\cdot)\)
\(\chi_{290}(187,\cdot)\)
\(\chi_{290}(207,\cdot)\)
\(\chi_{290}(237,\cdot)\)
\(\chi_{290}(267,\cdot)\)
\(\chi_{290}(283,\cdot)\)
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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 ]
\((117,31)\) → \((-i,e\left(\frac{11}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 290 }(63, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(i\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
