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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1058, base_ring=CyclotomicField(506))
M = H._module
chi = DirichletCharacter(H, M([424]))
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pari:[g,chi] = znchar(Mod(31,1058))
\(\chi_{1058}(3,\cdot)\)
\(\chi_{1058}(9,\cdot)\)
\(\chi_{1058}(13,\cdot)\)
\(\chi_{1058}(25,\cdot)\)
\(\chi_{1058}(27,\cdot)\)
\(\chi_{1058}(29,\cdot)\)
\(\chi_{1058}(31,\cdot)\)
\(\chi_{1058}(35,\cdot)\)
\(\chi_{1058}(39,\cdot)\)
\(\chi_{1058}(41,\cdot)\)
\(\chi_{1058}(49,\cdot)\)
\(\chi_{1058}(55,\cdot)\)
\(\chi_{1058}(59,\cdot)\)
\(\chi_{1058}(71,\cdot)\)
\(\chi_{1058}(73,\cdot)\)
\(\chi_{1058}(75,\cdot)\)
\(\chi_{1058}(77,\cdot)\)
\(\chi_{1058}(81,\cdot)\)
\(\chi_{1058}(85,\cdot)\)
\(\chi_{1058}(87,\cdot)\)
\(\chi_{1058}(95,\cdot)\)
\(\chi_{1058}(101,\cdot)\)
\(\chi_{1058}(105,\cdot)\)
\(\chi_{1058}(117,\cdot)\)
\(\chi_{1058}(119,\cdot)\)
\(\chi_{1058}(121,\cdot)\)
\(\chi_{1058}(123,\cdot)\)
\(\chi_{1058}(127,\cdot)\)
\(\chi_{1058}(131,\cdot)\)
\(\chi_{1058}(133,\cdot)\)
...
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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 ]
\(5\) → \(e\left(\frac{212}{253}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1058 }(31, a) \) |
\(1\) | \(1\) | \(e\left(\frac{103}{253}\right)\) | \(e\left(\frac{212}{253}\right)\) | \(e\left(\frac{24}{253}\right)\) | \(e\left(\frac{206}{253}\right)\) | \(e\left(\frac{60}{253}\right)\) | \(e\left(\frac{53}{253}\right)\) | \(e\left(\frac{62}{253}\right)\) | \(e\left(\frac{76}{253}\right)\) | \(e\left(\frac{34}{253}\right)\) | \(e\left(\frac{127}{253}\right)\) |
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sage:chi.jacobi_sum(n)
