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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2032, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,0,11]))
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pari:[g,chi] = znchar(Mod(1375,2032))
\(\chi_{2032}(1167,\cdot)\)
\(\chi_{2032}(1375,\cdot)\)
\(\chi_{2032}(1487,\cdot)\)
\(\chi_{2032}(1583,\cdot)\)
\(\chi_{2032}(1599,\cdot)\)
\(\chi_{2032}(1679,\cdot)\)
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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 ]
\((255,1525,257)\) → \((-1,1,e\left(\frac{11}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2032 }(1375, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) |
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sage:chi.jacobi_sum(n)
