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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1170, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([10,0,1]))
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pari:[g,chi] = znchar(Mod(41,1170))
\(\chi_{1170}(41,\cdot)\)
\(\chi_{1170}(371,\cdot)\)
\(\chi_{1170}(401,\cdot)\)
\(\chi_{1170}(461,\cdot)\)
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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 ]
\((911,937,1081)\) → \((e\left(\frac{5}{6}\right),1,e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1170 }(41, a) \) |
\(1\) | \(1\) | \(i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(-1\) |
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sage:chi.jacobi_sum(n)
