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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12138, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,17,20]))
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pari:[g,chi] = znchar(Mod(545,12138))
\(\chi_{12138}(545,\cdot)\)
\(\chi_{12138}(1259,\cdot)\)
\(\chi_{12138}(1973,\cdot)\)
\(\chi_{12138}(2687,\cdot)\)
\(\chi_{12138}(3401,\cdot)\)
\(\chi_{12138}(4115,\cdot)\)
\(\chi_{12138}(4829,\cdot)\)
\(\chi_{12138}(5543,\cdot)\)
\(\chi_{12138}(6257,\cdot)\)
\(\chi_{12138}(6971,\cdot)\)
\(\chi_{12138}(7685,\cdot)\)
\(\chi_{12138}(8399,\cdot)\)
\(\chi_{12138}(9113,\cdot)\)
\(\chi_{12138}(10541,\cdot)\)
\(\chi_{12138}(11255,\cdot)\)
\(\chi_{12138}(11969,\cdot)\)
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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 ]
\((8093,10405,9829)\) → \((-1,-1,e\left(\frac{10}{17}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 12138 }(545, a) \) |
\(1\) | \(1\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) |
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sage:chi.jacobi_sum(n)
