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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9386, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([11,6]))
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pari:[g,chi] = znchar(Mod(2165,9386))
\(\chi_{9386}(721,\cdot)\)
\(\chi_{9386}(2165,\cdot)\)
\(\chi_{9386}(4331,\cdot)\)
\(\chi_{9386}(7941,\cdot)\)
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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 ]
\((1445,3251)\) → \((e\left(\frac{11}{12}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 9386 }(2165, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(i\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
