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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1976, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,0,6,11]))
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pari:[g,chi] = znchar(Mod(471,1976))
\(\chi_{1976}(471,\cdot)\)
\(\chi_{1976}(887,\cdot)\)
\(\chi_{1976}(1439,\cdot)\)
\(\chi_{1976}(1511,\cdot)\)
\(\chi_{1976}(1751,\cdot)\)
\(\chi_{1976}(1959,\cdot)\)
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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 ]
\((495,989,457,1769)\) → \((-1,1,e\left(\frac{1}{3}\right),e\left(\frac{11}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 1976 }(471, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) |
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sage:chi.jacobi_sum(n)
