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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4641, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,8,0,9]))
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pari:[g,chi] = znchar(Mod(1171,4641))
\(\chi_{4641}(508,\cdot)\)
\(\chi_{4641}(1171,\cdot)\)
\(\chi_{4641}(1600,\cdot)\)
\(\chi_{4641}(2263,\cdot)\)
\(\chi_{4641}(3238,\cdot)\)
\(\chi_{4641}(3511,\cdot)\)
\(\chi_{4641}(3901,\cdot)\)
\(\chi_{4641}(4174,\cdot)\)
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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 ]
\((3095,3979,3928,547)\) → \((1,e\left(\frac{1}{3}\right),1,e\left(\frac{3}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(19\) | \(20\) | \(22\) |
| \( \chi_{ 4641 }(1171, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(-i\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) |
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sage:chi.jacobi_sum(n)
