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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,39,10]))
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pari:[g,chi] = znchar(Mod(1417,3150))
\(\chi_{3150}(187,\cdot)\)
\(\chi_{3150}(283,\cdot)\)
\(\chi_{3150}(313,\cdot)\)
\(\chi_{3150}(787,\cdot)\)
\(\chi_{3150}(817,\cdot)\)
\(\chi_{3150}(913,\cdot)\)
\(\chi_{3150}(1417,\cdot)\)
\(\chi_{3150}(1447,\cdot)\)
\(\chi_{3150}(1573,\cdot)\)
\(\chi_{3150}(2047,\cdot)\)
\(\chi_{3150}(2077,\cdot)\)
\(\chi_{3150}(2173,\cdot)\)
\(\chi_{3150}(2203,\cdot)\)
\(\chi_{3150}(2677,\cdot)\)
\(\chi_{3150}(2803,\cdot)\)
\(\chi_{3150}(2833,\cdot)\)
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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 ]
\((2801,127,451)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{13}{20}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3150 }(1417, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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sage:chi.jacobi_sum(n)
