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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(1932))
M = H._module
chi = DirichletCharacter(H, M([966,1725,952]))
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pari:[g,chi] = znchar(Mod(69,40310))
\(\chi_{40310}(69,\cdot)\)
\(\chi_{40310}(89,\cdot)\)
\(\chi_{40310}(159,\cdot)\)
\(\chi_{40310}(259,\cdot)\)
\(\chi_{40310}(309,\cdot)\)
\(\chi_{40310}(329,\cdot)\)
\(\chi_{40310}(359,\cdot)\)
\(\chi_{40310}(569,\cdot)\)
\(\chi_{40310}(669,\cdot)\)
\(\chi_{40310}(699,\cdot)\)
\(\chi_{40310}(839,\cdot)\)
\(\chi_{40310}(859,\cdot)\)
\(\chi_{40310}(989,\cdot)\)
\(\chi_{40310}(1059,\cdot)\)
\(\chi_{40310}(1149,\cdot)\)
\(\chi_{40310}(1179,\cdot)\)
\(\chi_{40310}(1229,\cdot)\)
\(\chi_{40310}(1239,\cdot)\)
\(\chi_{40310}(1249,\cdot)\)
\(\chi_{40310}(1279,\cdot)\)
\(\chi_{40310}(1419,\cdot)\)
\(\chi_{40310}(1439,\cdot)\)
\(\chi_{40310}(1489,\cdot)\)
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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 ]
\((24187,19461,16821)\) → \((-1,e\left(\frac{25}{28}\right),e\left(\frac{34}{69}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(69, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{323}{1932}\right)\) | \(e\left(\frac{823}{966}\right)\) | \(e\left(\frac{323}{966}\right)\) | \(e\left(\frac{1489}{1932}\right)\) | \(e\left(\frac{52}{483}\right)\) | \(e\left(\frac{269}{276}\right)\) | \(e\left(\frac{181}{1932}\right)\) | \(e\left(\frac{37}{1932}\right)\) | \(e\left(\frac{213}{322}\right)\) | \(e\left(\frac{323}{644}\right)\) |
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sage:chi.jacobi_sum(n)
