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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2912, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,9,8,2]))
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pari:[g,chi] = znchar(Mod(2825,2912))
\(\chi_{2912}(569,\cdot)\)
\(\chi_{2912}(1369,\cdot)\)
\(\chi_{2912}(2025,\cdot)\)
\(\chi_{2912}(2825,\cdot)\)
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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 ]
\((2367,1093,1249,2017)\) → \((1,-i,e\left(\frac{2}{3}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 2912 }(2825, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
