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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(729, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([28]))
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pari:[g,chi] = znchar(Mod(217,729))
\(\chi_{729}(28,\cdot)\)
\(\chi_{729}(55,\cdot)\)
\(\chi_{729}(109,\cdot)\)
\(\chi_{729}(136,\cdot)\)
\(\chi_{729}(190,\cdot)\)
\(\chi_{729}(217,\cdot)\)
\(\chi_{729}(271,\cdot)\)
\(\chi_{729}(298,\cdot)\)
\(\chi_{729}(352,\cdot)\)
\(\chi_{729}(379,\cdot)\)
\(\chi_{729}(433,\cdot)\)
\(\chi_{729}(460,\cdot)\)
\(\chi_{729}(514,\cdot)\)
\(\chi_{729}(541,\cdot)\)
\(\chi_{729}(595,\cdot)\)
\(\chi_{729}(622,\cdot)\)
\(\chi_{729}(676,\cdot)\)
\(\chi_{729}(703,\cdot)\)
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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 ]
\(2\) → \(e\left(\frac{14}{27}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 729 }(217, a) \) |
\(1\) | \(1\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) |
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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)
