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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(411, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,20]))
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pari:[g,chi] = znchar(Mod(256,411))
\(\chi_{411}(16,\cdot)\)
\(\chi_{411}(34,\cdot)\)
\(\chi_{411}(73,\cdot)\)
\(\chi_{411}(88,\cdot)\)
\(\chi_{411}(115,\cdot)\)
\(\chi_{411}(133,\cdot)\)
\(\chi_{411}(175,\cdot)\)
\(\chi_{411}(187,\cdot)\)
\(\chi_{411}(193,\cdot)\)
\(\chi_{411}(196,\cdot)\)
\(\chi_{411}(211,\cdot)\)
\(\chi_{411}(256,\cdot)\)
\(\chi_{411}(259,\cdot)\)
\(\chi_{411}(334,\cdot)\)
\(\chi_{411}(346,\cdot)\)
\(\chi_{411}(397,\cdot)\)
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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 ]
\((275,277)\) → \((1,e\left(\frac{10}{17}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 411 }(256, a) \) |
\(1\) | \(1\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(1\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) |
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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)
