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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(538, base_ring=CyclotomicField(134))
M = H._module
chi = DirichletCharacter(H, M([75]))
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pari:[g,chi] = znchar(Mod(369,538))
\(\chi_{538}(9,\cdot)\)
\(\chi_{538}(11,\cdot)\)
\(\chi_{538}(13,\cdot)\)
\(\chi_{538}(43,\cdot)\)
\(\chi_{538}(45,\cdot)\)
\(\chi_{538}(49,\cdot)\)
\(\chi_{538}(51,\cdot)\)
\(\chi_{538}(55,\cdot)\)
\(\chi_{538}(65,\cdot)\)
\(\chi_{538}(73,\cdot)\)
\(\chi_{538}(79,\cdot)\)
\(\chi_{538}(89,\cdot)\)
\(\chi_{538}(97,\cdot)\)
\(\chi_{538}(103,\cdot)\)
\(\chi_{538}(127,\cdot)\)
\(\chi_{538}(133,\cdot)\)
\(\chi_{538}(149,\cdot)\)
\(\chi_{538}(151,\cdot)\)
\(\chi_{538}(189,\cdot)\)
\(\chi_{538}(191,\cdot)\)
\(\chi_{538}(199,\cdot)\)
\(\chi_{538}(203,\cdot)\)
\(\chi_{538}(207,\cdot)\)
\(\chi_{538}(211,\cdot)\)
\(\chi_{538}(215,\cdot)\)
\(\chi_{538}(217,\cdot)\)
\(\chi_{538}(225,\cdot)\)
\(\chi_{538}(231,\cdot)\)
\(\chi_{538}(233,\cdot)\)
\(\chi_{538}(245,\cdot)\)
...
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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 ]
\(271\) → \(e\left(\frac{75}{134}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 538 }(369, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{134}\right)\) | \(e\left(\frac{28}{67}\right)\) | \(e\left(\frac{85}{134}\right)\) | \(e\left(\frac{1}{67}\right)\) | \(e\left(\frac{49}{67}\right)\) | \(e\left(\frac{32}{67}\right)\) | \(e\left(\frac{57}{134}\right)\) | \(e\left(\frac{103}{134}\right)\) | \(e\left(\frac{109}{134}\right)\) | \(e\left(\frac{43}{67}\right)\) |
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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)
