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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,12]))
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pari:[g,chi] = znchar(Mod(331,363))
\(\chi_{363}(34,\cdot)\)
\(\chi_{363}(67,\cdot)\)
\(\chi_{363}(100,\cdot)\)
\(\chi_{363}(133,\cdot)\)
\(\chi_{363}(166,\cdot)\)
\(\chi_{363}(199,\cdot)\)
\(\chi_{363}(232,\cdot)\)
\(\chi_{363}(265,\cdot)\)
\(\chi_{363}(298,\cdot)\)
\(\chi_{363}(331,\cdot)\)
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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 ]
\((122,244)\) → \((1,e\left(\frac{6}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 363 }(331, a) \) |
\(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) |
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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)
