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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7581, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([0,19,101]))
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pari:[g,chi] = znchar(Mod(31,7581))
\(\chi_{7581}(31,\cdot)\)
\(\chi_{7581}(103,\cdot)\)
\(\chi_{7581}(502,\cdot)\)
\(\chi_{7581}(829,\cdot)\)
\(\chi_{7581}(901,\cdot)\)
\(\chi_{7581}(1228,\cdot)\)
\(\chi_{7581}(1300,\cdot)\)
\(\chi_{7581}(1627,\cdot)\)
\(\chi_{7581}(1699,\cdot)\)
\(\chi_{7581}(2026,\cdot)\)
\(\chi_{7581}(2425,\cdot)\)
\(\chi_{7581}(2497,\cdot)\)
\(\chi_{7581}(2824,\cdot)\)
\(\chi_{7581}(2896,\cdot)\)
\(\chi_{7581}(3223,\cdot)\)
\(\chi_{7581}(3295,\cdot)\)
\(\chi_{7581}(3622,\cdot)\)
\(\chi_{7581}(3694,\cdot)\)
\(\chi_{7581}(4021,\cdot)\)
\(\chi_{7581}(4093,\cdot)\)
\(\chi_{7581}(4420,\cdot)\)
\(\chi_{7581}(4492,\cdot)\)
\(\chi_{7581}(4819,\cdot)\)
\(\chi_{7581}(4891,\cdot)\)
\(\chi_{7581}(5218,\cdot)\)
\(\chi_{7581}(5290,\cdot)\)
\(\chi_{7581}(5617,\cdot)\)
\(\chi_{7581}(5689,\cdot)\)
\(\chi_{7581}(6016,\cdot)\)
\(\chi_{7581}(6088,\cdot)\)
...
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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 ]
\((2528,6499,1807)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{101}{114}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
| \( \chi_{ 7581 }(31, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{31}{38}\right)\) |
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sage:chi.jacobi_sum(n)
