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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1680, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,9,0,0,4]))
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pari:[g,chi] = znchar(Mod(541,1680))
\(\chi_{1680}(541,\cdot)\)
\(\chi_{1680}(781,\cdot)\)
\(\chi_{1680}(1381,\cdot)\)
\(\chi_{1680}(1621,\cdot)\)
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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 ]
\((1471,421,1121,337,241)\) → \((1,-i,1,1,e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1680 }(541, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-1\) | \(-i\) |
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sage:chi.jacobi_sum(n)
