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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(56644, base_ring=CyclotomicField(5712))
M = H._module
chi = DirichletCharacter(H, M([0,4216,4851]))
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pari:[g,chi] = znchar(Mod(45,56644))
\(\chi_{56644}(5,\cdot)\)
\(\chi_{56644}(45,\cdot)\)
\(\chi_{56644}(61,\cdot)\)
\(\chi_{56644}(73,\cdot)\)
\(\chi_{56644}(173,\cdot)\)
\(\chi_{56644}(201,\cdot)\)
\(\chi_{56644}(241,\cdot)\)
\(\chi_{56644}(269,\cdot)\)
\(\chi_{56644}(369,\cdot)\)
\(\chi_{56644}(381,\cdot)\)
\(\chi_{56644}(397,\cdot)\)
\(\chi_{56644}(437,\cdot)\)
\(\chi_{56644}(453,\cdot)\)
\(\chi_{56644}(465,\cdot)\)
\(\chi_{56644}(481,\cdot)\)
\(\chi_{56644}(537,\cdot)\)
\(\chi_{56644}(549,\cdot)\)
\(\chi_{56644}(605,\cdot)\)
\(\chi_{56644}(649,\cdot)\)
\(\chi_{56644}(677,\cdot)\)
\(\chi_{56644}(745,\cdot)\)
\(\chi_{56644}(789,\cdot)\)
\(\chi_{56644}(845,\cdot)\)
\(\chi_{56644}(857,\cdot)\)
\(\chi_{56644}(873,\cdot)\)
\(\chi_{56644}(929,\cdot)\)
\(\chi_{56644}(941,\cdot)\)
\(\chi_{56644}(957,\cdot)\)
\(\chi_{56644}(997,\cdot)\)
\(\chi_{56644}(1013,\cdot)\)
...
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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 ]
\((28323,50865,34105)\) → \((1,e\left(\frac{31}{42}\right),e\left(\frac{231}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 56644 }(45, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3355}{5712}\right)\) | \(e\left(\frac{5063}{5712}\right)\) | \(e\left(\frac{499}{2856}\right)\) | \(e\left(\frac{325}{5712}\right)\) | \(e\left(\frac{387}{476}\right)\) | \(e\left(\frac{451}{952}\right)\) | \(e\left(\frac{295}{408}\right)\) | \(e\left(\frac{2477}{5712}\right)\) | \(e\left(\frac{2207}{2856}\right)\) | \(e\left(\frac{1451}{1904}\right)\) |
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sage:chi.jacobi_sum(n)
