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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(891, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([5,27]))
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pari:[g,chi] = znchar(Mod(8,891))
\(\chi_{891}(8,\cdot)\)
\(\chi_{891}(17,\cdot)\)
\(\chi_{891}(35,\cdot)\)
\(\chi_{891}(62,\cdot)\)
\(\chi_{891}(116,\cdot)\)
\(\chi_{891}(206,\cdot)\)
\(\chi_{891}(233,\cdot)\)
\(\chi_{891}(260,\cdot)\)
\(\chi_{891}(305,\cdot)\)
\(\chi_{891}(314,\cdot)\)
\(\chi_{891}(332,\cdot)\)
\(\chi_{891}(359,\cdot)\)
\(\chi_{891}(413,\cdot)\)
\(\chi_{891}(503,\cdot)\)
\(\chi_{891}(530,\cdot)\)
\(\chi_{891}(557,\cdot)\)
\(\chi_{891}(602,\cdot)\)
\(\chi_{891}(611,\cdot)\)
\(\chi_{891}(629,\cdot)\)
\(\chi_{891}(656,\cdot)\)
\(\chi_{891}(710,\cdot)\)
\(\chi_{891}(800,\cdot)\)
\(\chi_{891}(827,\cdot)\)
\(\chi_{891}(854,\cdot)\)
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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 ]
\((650,244)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 891 }(8, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{31}{90}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) |
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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)
