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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(729, base_ring=CyclotomicField(162))
M = H._module
chi = DirichletCharacter(H, M([41]))
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pari:[g,chi] = znchar(Mod(287,729))
\(\chi_{729}(8,\cdot)\)
\(\chi_{729}(17,\cdot)\)
\(\chi_{729}(35,\cdot)\)
\(\chi_{729}(44,\cdot)\)
\(\chi_{729}(62,\cdot)\)
\(\chi_{729}(71,\cdot)\)
\(\chi_{729}(89,\cdot)\)
\(\chi_{729}(98,\cdot)\)
\(\chi_{729}(116,\cdot)\)
\(\chi_{729}(125,\cdot)\)
\(\chi_{729}(143,\cdot)\)
\(\chi_{729}(152,\cdot)\)
\(\chi_{729}(170,\cdot)\)
\(\chi_{729}(179,\cdot)\)
\(\chi_{729}(197,\cdot)\)
\(\chi_{729}(206,\cdot)\)
\(\chi_{729}(224,\cdot)\)
\(\chi_{729}(233,\cdot)\)
\(\chi_{729}(251,\cdot)\)
\(\chi_{729}(260,\cdot)\)
\(\chi_{729}(278,\cdot)\)
\(\chi_{729}(287,\cdot)\)
\(\chi_{729}(305,\cdot)\)
\(\chi_{729}(314,\cdot)\)
\(\chi_{729}(332,\cdot)\)
\(\chi_{729}(341,\cdot)\)
\(\chi_{729}(359,\cdot)\)
\(\chi_{729}(368,\cdot)\)
\(\chi_{729}(386,\cdot)\)
\(\chi_{729}(395,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{41}{162}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 729 }(287, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{162}\right)\) | \(e\left(\frac{41}{81}\right)\) | \(e\left(\frac{133}{162}\right)\) | \(e\left(\frac{58}{81}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{101}{162}\right)\) | \(e\left(\frac{2}{81}\right)\) | \(e\left(\frac{157}{162}\right)\) | \(e\left(\frac{1}{81}\right)\) |
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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)
