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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28665, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([0,63,76,35]))
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pari:[g,chi] = znchar(Mod(25993,28665))
\(\chi_{28665}(613,\cdot)\)
\(\chi_{28665}(982,\cdot)\)
\(\chi_{28665}(1423,\cdot)\)
\(\chi_{28665}(2377,\cdot)\)
\(\chi_{28665}(4708,\cdot)\)
\(\chi_{28665}(6472,\cdot)\)
\(\chi_{28665}(8803,\cdot)\)
\(\chi_{28665}(9172,\cdot)\)
\(\chi_{28665}(9613,\cdot)\)
\(\chi_{28665}(10567,\cdot)\)
\(\chi_{28665}(12898,\cdot)\)
\(\chi_{28665}(13267,\cdot)\)
\(\chi_{28665}(13708,\cdot)\)
\(\chi_{28665}(14662,\cdot)\)
\(\chi_{28665}(16993,\cdot)\)
\(\chi_{28665}(17362,\cdot)\)
\(\chi_{28665}(17803,\cdot)\)
\(\chi_{28665}(18757,\cdot)\)
\(\chi_{28665}(21457,\cdot)\)
\(\chi_{28665}(21898,\cdot)\)
\(\chi_{28665}(25183,\cdot)\)
\(\chi_{28665}(25552,\cdot)\)
\(\chi_{28665}(25993,\cdot)\)
\(\chi_{28665}(26947,\cdot)\)
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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 ]
\((25481,11467,18721,11026)\) → \((1,-i,e\left(\frac{19}{21}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) | \(29\) |
| \( \chi_{ 28665 }(25993, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(i\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) |
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sage:chi.jacobi_sum(n)
