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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6660, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,9,0,10]))
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pari:[g,chi] = znchar(Mod(2771,6660))
\(\chi_{6660}(71,\cdot)\)
\(\chi_{6660}(971,\cdot)\)
\(\chi_{6660}(2051,\cdot)\)
\(\chi_{6660}(2771,\cdot)\)
\(\chi_{6660}(6191,\cdot)\)
\(\chi_{6660}(6371,\cdot)\)
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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 ]
\((3331,3701,3997,3961)\) → \((-1,-1,1,e\left(\frac{5}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \( \chi_{ 6660 }(2771, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{11}{18}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
