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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,0,1]))
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pari:[g,chi] = znchar(Mod(914,1045))
\(\chi_{1045}(34,\cdot)\)
\(\chi_{1045}(89,\cdot)\)
\(\chi_{1045}(364,\cdot)\)
\(\chi_{1045}(584,\cdot)\)
\(\chi_{1045}(694,\cdot)\)
\(\chi_{1045}(914,\cdot)\)
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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 ]
\((837,761,496)\) → \((-1,1,e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 1045 }(914, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) |
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sage:chi.jacobi_sum(n)
