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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(592, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,25]))
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pari:[g,chi] = znchar(Mod(279,592))
\(\chi_{592}(39,\cdot)\)
\(\chi_{592}(55,\cdot)\)
\(\chi_{592}(87,\cdot)\)
\(\chi_{592}(135,\cdot)\)
\(\chi_{592}(167,\cdot)\)
\(\chi_{592}(183,\cdot)\)
\(\chi_{592}(279,\cdot)\)
\(\chi_{592}(311,\cdot)\)
\(\chi_{592}(375,\cdot)\)
\(\chi_{592}(439,\cdot)\)
\(\chi_{592}(503,\cdot)\)
\(\chi_{592}(535,\cdot)\)
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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 ]
\((223,149,113)\) → \((-1,-1,e\left(\frac{25}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 592 }(279, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) |
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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)
