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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6336, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([0,105,40,72]))
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pari:[g,chi] = znchar(Mod(4585,6336))
\(\chi_{6336}(25,\cdot)\)
\(\chi_{6336}(169,\cdot)\)
\(\chi_{6336}(313,\cdot)\)
\(\chi_{6336}(553,\cdot)\)
\(\chi_{6336}(697,\cdot)\)
\(\chi_{6336}(841,\cdot)\)
\(\chi_{6336}(889,\cdot)\)
\(\chi_{6336}(1417,\cdot)\)
\(\chi_{6336}(1609,\cdot)\)
\(\chi_{6336}(1753,\cdot)\)
\(\chi_{6336}(1897,\cdot)\)
\(\chi_{6336}(2137,\cdot)\)
\(\chi_{6336}(2281,\cdot)\)
\(\chi_{6336}(2425,\cdot)\)
\(\chi_{6336}(2473,\cdot)\)
\(\chi_{6336}(3001,\cdot)\)
\(\chi_{6336}(3193,\cdot)\)
\(\chi_{6336}(3337,\cdot)\)
\(\chi_{6336}(3481,\cdot)\)
\(\chi_{6336}(3721,\cdot)\)
\(\chi_{6336}(3865,\cdot)\)
\(\chi_{6336}(4009,\cdot)\)
\(\chi_{6336}(4057,\cdot)\)
\(\chi_{6336}(4585,\cdot)\)
\(\chi_{6336}(4777,\cdot)\)
\(\chi_{6336}(4921,\cdot)\)
\(\chi_{6336}(5065,\cdot)\)
\(\chi_{6336}(5305,\cdot)\)
\(\chi_{6336}(5449,\cdot)\)
\(\chi_{6336}(5593,\cdot)\)
...
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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 ]
\((4159,4357,3521,1729)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{1}{3}\right),e\left(\frac{3}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 6336 }(4585, a) \) |
\(1\) | \(1\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{9}{40}\right)\) |
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sage:chi.jacobi_sum(n)
