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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4563, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([14,3]))
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pari:[g,chi] = znchar(Mod(2344,4563))
\(\chi_{4563}(823,\cdot)\)
\(\chi_{4563}(1375,\cdot)\)
\(\chi_{4563}(2344,\cdot)\)
\(\chi_{4563}(2896,\cdot)\)
\(\chi_{4563}(3865,\cdot)\)
\(\chi_{4563}(4417,\cdot)\)
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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 ]
\((677,3889)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 4563 }(2344, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
