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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(42237, base_ring=CyclotomicField(6))
M = H._module
chi = DirichletCharacter(H, M([4,1,0]))
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pari:[g,chi] = znchar(Mod(18412,42237))
\(\chi_{42237}(18412,\cdot)\)
\(\chi_{42237}(27076,\cdot)\)
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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 ]
\((32852,38989,12637)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{6}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 42237 }(18412, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
