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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4320, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,9,10,9]))
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pari:[g,chi] = znchar(Mod(79,4320))
\(\chi_{4320}(79,\cdot)\)
\(\chi_{4320}(1039,\cdot)\)
\(\chi_{4320}(1519,\cdot)\)
\(\chi_{4320}(2479,\cdot)\)
\(\chi_{4320}(2959,\cdot)\)
\(\chi_{4320}(3919,\cdot)\)
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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 ]
\((2431,3781,2081,3457)\) → \((-1,-1,e\left(\frac{5}{9}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4320 }(79, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) |
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sage:chi.jacobi_sum(n)
