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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1083, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,8]))
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pari:[g,chi] = znchar(Mod(229,1083))
\(\chi_{1083}(58,\cdot)\)
\(\chi_{1083}(115,\cdot)\)
\(\chi_{1083}(172,\cdot)\)
\(\chi_{1083}(229,\cdot)\)
\(\chi_{1083}(286,\cdot)\)
\(\chi_{1083}(343,\cdot)\)
\(\chi_{1083}(400,\cdot)\)
\(\chi_{1083}(457,\cdot)\)
\(\chi_{1083}(514,\cdot)\)
\(\chi_{1083}(571,\cdot)\)
\(\chi_{1083}(628,\cdot)\)
\(\chi_{1083}(685,\cdot)\)
\(\chi_{1083}(742,\cdot)\)
\(\chi_{1083}(799,\cdot)\)
\(\chi_{1083}(856,\cdot)\)
\(\chi_{1083}(913,\cdot)\)
\(\chi_{1083}(970,\cdot)\)
\(\chi_{1083}(1027,\cdot)\)
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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 ]
\((362,724)\) → \((1,e\left(\frac{4}{19}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1083 }(229, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) |
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sage:chi.jacobi_sum(n)
