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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3825, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([8,6,3]))
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pari:[g,chi] = znchar(Mod(574,3825))
\(\chi_{3825}(574,\cdot)\)
\(\chi_{3825}(1024,\cdot)\)
\(\chi_{3825}(1849,\cdot)\)
\(\chi_{3825}(2299,\cdot)\)
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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 ]
\((2126,2602,2026)\) → \((e\left(\frac{2}{3}\right),-1,i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(19\) | \(22\) |
| \( \chi_{ 3825 }(574, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) |
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sage:chi.jacobi_sum(n)
