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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3825, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([20,24,0]))
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pari:[g,chi] = znchar(Mod(511,3825))
\(\chi_{3825}(256,\cdot)\)
\(\chi_{3825}(511,\cdot)\)
\(\chi_{3825}(1021,\cdot)\)
\(\chi_{3825}(1786,\cdot)\)
\(\chi_{3825}(2041,\cdot)\)
\(\chi_{3825}(2806,\cdot)\)
\(\chi_{3825}(3316,\cdot)\)
\(\chi_{3825}(3571,\cdot)\)
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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 ]
\((2126,2602,2026)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{4}{5}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(19\) | \(22\) |
| \( \chi_{ 3825 }(511, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) |
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sage:chi.jacobi_sum(n)
