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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2565, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,9,8]))
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pari:[g,chi] = znchar(Mod(674,2565))
\(\chi_{2565}(404,\cdot)\)
\(\chi_{2565}(674,\cdot)\)
\(\chi_{2565}(1214,\cdot)\)
\(\chi_{2565}(1619,\cdot)\)
\(\chi_{2565}(1754,\cdot)\)
\(\chi_{2565}(2429,\cdot)\)
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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 ]
\((191,1027,1351)\) → \((-1,-1,e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 2565 }(674, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) |
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sage:chi.jacobi_sum(n)
