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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(644))
M = H._module
chi = DirichletCharacter(H, M([161,207,588]))
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pari:[g,chi] = znchar(Mod(77,40310))
\(\chi_{40310}(77,\cdot)\)
\(\chi_{40310}(403,\cdot)\)
\(\chi_{40310}(453,\cdot)\)
\(\chi_{40310}(1007,\cdot)\)
\(\chi_{40310}(1017,\cdot)\)
\(\chi_{40310}(1237,\cdot)\)
\(\chi_{40310}(1287,\cdot)\)
\(\chi_{40310}(1303,\cdot)\)
\(\chi_{40310}(1563,\cdot)\)
\(\chi_{40310}(1593,\cdot)\)
\(\chi_{40310}(1813,\cdot)\)
\(\chi_{40310}(1887,\cdot)\)
\(\chi_{40310}(2397,\cdot)\)
\(\chi_{40310}(2463,\cdot)\)
\(\chi_{40310}(2747,\cdot)\)
\(\chi_{40310}(2753,\cdot)\)
\(\chi_{40310}(2757,\cdot)\)
\(\chi_{40310}(3113,\cdot)\)
\(\chi_{40310}(3303,\cdot)\)
\(\chi_{40310}(3693,\cdot)\)
\(\chi_{40310}(3947,\cdot)\)
\(\chi_{40310}(3983,\cdot)\)
\(\chi_{40310}(4137,\cdot)\)
\(\chi_{40310}(4493,\cdot)\)
\(\chi_{40310}(4503,\cdot)\)
\(\chi_{40310}(4513,\cdot)\)
\(\chi_{40310}(4527,\cdot)\)
\(\chi_{40310}(4783,\cdot)\)
\(\chi_{40310}(4803,\cdot)\)
\(\chi_{40310}(4817,\cdot)\)
...
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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 ]
\((24187,19461,16821)\) → \((i,e\left(\frac{9}{28}\right),e\left(\frac{21}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(77, a) \) |
\(1\) | \(1\) | \(e\left(\frac{255}{322}\right)\) | \(e\left(\frac{489}{644}\right)\) | \(e\left(\frac{94}{161}\right)\) | \(e\left(\frac{275}{644}\right)\) | \(e\left(\frac{625}{644}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{57}{644}\right)\) | \(e\left(\frac{355}{644}\right)\) | \(e\left(\frac{535}{644}\right)\) | \(e\left(\frac{121}{322}\right)\) |
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sage:chi.jacobi_sum(n)
