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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4560, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,6,0,9,2]))
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pari:[g,chi] = znchar(Mod(103,4560))
\(\chi_{4560}(103,\cdot)\)
\(\chi_{4560}(487,\cdot)\)
\(\chi_{4560}(1927,\cdot)\)
\(\chi_{4560}(3223,\cdot)\)
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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 ]
\((1711,1141,3041,2737,1921)\) → \((-1,-1,1,-i,e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4560 }(103, a) \) |
\(-1\) | \(1\) | \(i\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) |
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sage:chi.jacobi_sum(n)
