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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1870, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,56,75]))
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pari:[g,chi] = znchar(Mod(1349,1870))
\(\chi_{1870}(29,\cdot)\)
\(\chi_{1870}(39,\cdot)\)
\(\chi_{1870}(79,\cdot)\)
\(\chi_{1870}(129,\cdot)\)
\(\chi_{1870}(139,\cdot)\)
\(\chi_{1870}(249,\cdot)\)
\(\chi_{1870}(299,\cdot)\)
\(\chi_{1870}(369,\cdot)\)
\(\chi_{1870}(469,\cdot)\)
\(\chi_{1870}(479,\cdot)\)
\(\chi_{1870}(589,\cdot)\)
\(\chi_{1870}(789,\cdot)\)
\(\chi_{1870}(809,\cdot)\)
\(\chi_{1870}(959,\cdot)\)
\(\chi_{1870}(1009,\cdot)\)
\(\chi_{1870}(1119,\cdot)\)
\(\chi_{1870}(1129,\cdot)\)
\(\chi_{1870}(1179,\cdot)\)
\(\chi_{1870}(1229,\cdot)\)
\(\chi_{1870}(1289,\cdot)\)
\(\chi_{1870}(1349,\cdot)\)
\(\chi_{1870}(1399,\cdot)\)
\(\chi_{1870}(1459,\cdot)\)
\(\chi_{1870}(1469,\cdot)\)
\(\chi_{1870}(1559,\cdot)\)
\(\chi_{1870}(1569,\cdot)\)
\(\chi_{1870}(1669,\cdot)\)
\(\chi_{1870}(1689,\cdot)\)
\(\chi_{1870}(1729,\cdot)\)
\(\chi_{1870}(1779,\cdot)\)
...
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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 ]
\((1497,1531,1431)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{15}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 1870 }(1349, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(-i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) |
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sage:chi.jacobi_sum(n)
