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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([14]))
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pari:[g,chi] = znchar(Mod(685,729))
\(\chi_{729}(10,\cdot)\)
\(\chi_{729}(19,\cdot)\)
\(\chi_{729}(37,\cdot)\)
\(\chi_{729}(46,\cdot)\)
\(\chi_{729}(64,\cdot)\)
\(\chi_{729}(73,\cdot)\)
\(\chi_{729}(91,\cdot)\)
\(\chi_{729}(100,\cdot)\)
\(\chi_{729}(118,\cdot)\)
\(\chi_{729}(127,\cdot)\)
\(\chi_{729}(145,\cdot)\)
\(\chi_{729}(154,\cdot)\)
\(\chi_{729}(172,\cdot)\)
\(\chi_{729}(181,\cdot)\)
\(\chi_{729}(199,\cdot)\)
\(\chi_{729}(208,\cdot)\)
\(\chi_{729}(226,\cdot)\)
\(\chi_{729}(235,\cdot)\)
\(\chi_{729}(253,\cdot)\)
\(\chi_{729}(262,\cdot)\)
\(\chi_{729}(280,\cdot)\)
\(\chi_{729}(289,\cdot)\)
\(\chi_{729}(307,\cdot)\)
\(\chi_{729}(316,\cdot)\)
\(\chi_{729}(334,\cdot)\)
\(\chi_{729}(343,\cdot)\)
\(\chi_{729}(361,\cdot)\)
\(\chi_{729}(370,\cdot)\)
\(\chi_{729}(388,\cdot)\)
\(\chi_{729}(397,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{7}{81}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 729 }(685, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{81}\right)\) | \(e\left(\frac{14}{81}\right)\) | \(e\left(\frac{80}{81}\right)\) | \(e\left(\frac{4}{81}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{37}{81}\right)\) | \(e\left(\frac{56}{81}\right)\) | \(e\left(\frac{11}{81}\right)\) | \(e\left(\frac{28}{81}\right)\) |
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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)
