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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3234, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,10,9]))
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pari:[g,chi] = znchar(Mod(569,3234))
\(\chi_{3234}(557,\cdot)\)
\(\chi_{3234}(569,\cdot)\)
\(\chi_{3234}(1157,\cdot)\)
\(\chi_{3234}(1733,\cdot)\)
\(\chi_{3234}(1745,\cdot)\)
\(\chi_{3234}(2615,\cdot)\)
\(\chi_{3234}(2921,\cdot)\)
\(\chi_{3234}(3203,\cdot)\)
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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 ]
\((1079,199,2059)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3234 }(569, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) |
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sage:chi.jacobi_sum(n)
