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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8550, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,3,10]))
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pari:[g,chi] = znchar(Mod(7627,8550))
\(\chi_{8550}(103,\cdot)\)
\(\chi_{8550}(787,\cdot)\)
\(\chi_{8550}(1627,\cdot)\)
\(\chi_{8550}(1813,\cdot)\)
\(\chi_{8550}(2497,\cdot)\)
\(\chi_{8550}(2653,\cdot)\)
\(\chi_{8550}(3337,\cdot)\)
\(\chi_{8550}(3523,\cdot)\)
\(\chi_{8550}(4363,\cdot)\)
\(\chi_{8550}(5047,\cdot)\)
\(\chi_{8550}(5233,\cdot)\)
\(\chi_{8550}(5917,\cdot)\)
\(\chi_{8550}(6073,\cdot)\)
\(\chi_{8550}(7627,\cdot)\)
\(\chi_{8550}(7783,\cdot)\)
\(\chi_{8550}(8467,\cdot)\)
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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 ]
\((1901,1027,1351)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{20}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8550 }(7627, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
