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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28665, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([28,0,8,28]))
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pari:[g,chi] = znchar(Mod(12931,28665))
\(\chi_{28665}(646,\cdot)\)
\(\chi_{28665}(3721,\cdot)\)
\(\chi_{28665}(4741,\cdot)\)
\(\chi_{28665}(7816,\cdot)\)
\(\chi_{28665}(8836,\cdot)\)
\(\chi_{28665}(11911,\cdot)\)
\(\chi_{28665}(12931,\cdot)\)
\(\chi_{28665}(16006,\cdot)\)
\(\chi_{28665}(17026,\cdot)\)
\(\chi_{28665}(20101,\cdot)\)
\(\chi_{28665}(21121,\cdot)\)
\(\chi_{28665}(24196,\cdot)\)
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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 ]
\((25481,11467,18721,11026)\) → \((e\left(\frac{2}{3}\right),1,e\left(\frac{4}{21}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) | \(29\) |
| \( \chi_{ 28665 }(12931, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) |
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sage:chi.jacobi_sum(n)
