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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7581, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,3,7]))
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pari:[g,chi] = znchar(Mod(6436,7581))
\(\chi_{7581}(1921,\cdot)\)
\(\chi_{7581}(5542,\cdot)\)
\(\chi_{7581}(5722,\cdot)\)
\(\chi_{7581}(6109,\cdot)\)
\(\chi_{7581}(6436,\cdot)\)
\(\chi_{7581}(6760,\cdot)\)
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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 ]
\((2528,6499,1807)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{7}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
| \( \chi_{ 7581 }(6436, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
