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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8007, base_ring=CyclotomicField(208))
M = H._module
chi = DirichletCharacter(H, M([0,65,116]))
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pari:[g,chi] = znchar(Mod(583,8007))
\(\chi_{8007}(316,\cdot)\)
\(\chi_{8007}(337,\cdot)\)
\(\chi_{8007}(346,\cdot)\)
\(\chi_{8007}(379,\cdot)\)
\(\chi_{8007}(430,\cdot)\)
\(\chi_{8007}(439,\cdot)\)
\(\chi_{8007}(448,\cdot)\)
\(\chi_{8007}(583,\cdot)\)
\(\chi_{8007}(673,\cdot)\)
\(\chi_{8007}(826,\cdot)\)
\(\chi_{8007}(844,\cdot)\)
\(\chi_{8007}(877,\cdot)\)
\(\chi_{8007}(913,\cdot)\)
\(\chi_{8007}(940,\cdot)\)
\(\chi_{8007}(949,\cdot)\)
\(\chi_{8007}(1153,\cdot)\)
\(\chi_{8007}(1285,\cdot)\)
\(\chi_{8007}(1315,\cdot)\)
\(\chi_{8007}(1321,\cdot)\)
\(\chi_{8007}(1372,\cdot)\)
\(\chi_{8007}(1405,\cdot)\)
\(\chi_{8007}(1525,\cdot)\)
\(\chi_{8007}(1876,\cdot)\)
\(\chi_{8007}(1882,\cdot)\)
\(\chi_{8007}(2119,\cdot)\)
\(\chi_{8007}(2200,\cdot)\)
\(\chi_{8007}(2326,\cdot)\)
\(\chi_{8007}(2434,\cdot)\)
\(\chi_{8007}(2458,\cdot)\)
\(\chi_{8007}(2557,\cdot)\)
...
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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 ]
\((5339,1414,7855)\) → \((1,e\left(\frac{5}{16}\right),e\left(\frac{29}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 8007 }(583, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{104}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{25}{208}\right)\) | \(e\left(\frac{87}{208}\right)\) | \(e\left(\frac{3}{104}\right)\) | \(e\left(\frac{27}{208}\right)\) | \(e\left(\frac{167}{208}\right)\) | \(-i\) | \(e\left(\frac{89}{208}\right)\) | \(e\left(\frac{1}{26}\right)\) |
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sage:chi.jacobi_sum(n)
