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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28665, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([10,9,10,7]))
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pari:[g,chi] = znchar(Mod(2273,28665))
\(\chi_{28665}(227,\cdot)\)
\(\chi_{28665}(2273,\cdot)\)
\(\chi_{28665}(6242,\cdot)\)
\(\chi_{28665}(22718,\cdot)\)
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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 ]
\((25481,11467,18721,11026)\) → \((e\left(\frac{5}{6}\right),-i,e\left(\frac{5}{6}\right),e\left(\frac{7}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) | \(29\) |
| \( \chi_{ 28665 }(2273, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) |
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sage:chi.jacobi_sum(n)
