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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,9,8]))
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pari:[g,chi] = znchar(Mod(265,576))
\(\chi_{576}(25,\cdot)\)
\(\chi_{576}(121,\cdot)\)
\(\chi_{576}(169,\cdot)\)
\(\chi_{576}(265,\cdot)\)
\(\chi_{576}(313,\cdot)\)
\(\chi_{576}(409,\cdot)\)
\(\chi_{576}(457,\cdot)\)
\(\chi_{576}(553,\cdot)\)
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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 ]
\((127,325,65)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 576 }(265, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) |
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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)
