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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2565, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([8,3,4]))
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pari:[g,chi] = znchar(Mod(2287,2565))
\(\chi_{2565}(748,\cdot)\)
\(\chi_{2565}(847,\cdot)\)
\(\chi_{2565}(1873,\cdot)\)
\(\chi_{2565}(2287,\cdot)\)
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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 ]
\((191,1027,1351)\) → \((e\left(\frac{2}{3}\right),i,e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 2565 }(2287, a) \) |
\(-1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(e\left(\frac{2}{3}\right)\) | \(-i\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) |
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sage:chi.jacobi_sum(n)
