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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([0,0,22,35]))
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pari:[g,chi] = znchar(Mod(4456,28665))
\(\chi_{28665}(1486,\cdot)\)
\(\chi_{28665}(4456,\cdot)\)
\(\chi_{28665}(5581,\cdot)\)
\(\chi_{28665}(8551,\cdot)\)
\(\chi_{28665}(9676,\cdot)\)
\(\chi_{28665}(12646,\cdot)\)
\(\chi_{28665}(13771,\cdot)\)
\(\chi_{28665}(16741,\cdot)\)
\(\chi_{28665}(20836,\cdot)\)
\(\chi_{28665}(21961,\cdot)\)
\(\chi_{28665}(24931,\cdot)\)
\(\chi_{28665}(26056,\cdot)\)
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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 ]
\((25481,11467,18721,11026)\) → \((1,1,e\left(\frac{11}{21}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) | \(29\) |
| \( \chi_{ 28665 }(4456, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(-1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) |
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sage:chi.jacobi_sum(n)
