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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([966,621,854]))
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pari:[g,chi] = znchar(Mod(19,40310))
\(\chi_{40310}(19,\cdot)\)
\(\chi_{40310}(119,\cdot)\)
\(\chi_{40310}(189,\cdot)\)
\(\chi_{40310}(229,\cdot)\)
\(\chi_{40310}(269,\cdot)\)
\(\chi_{40310}(379,\cdot)\)
\(\chi_{40310}(449,\cdot)\)
\(\chi_{40310}(519,\cdot)\)
\(\chi_{40310}(549,\cdot)\)
\(\chi_{40310}(559,\cdot)\)
\(\chi_{40310}(649,\cdot)\)
\(\chi_{40310}(809,\cdot)\)
\(\chi_{40310}(849,\cdot)\)
\(\chi_{40310}(949,\cdot)\)
\(\chi_{40310}(1029,\cdot)\)
\(\chi_{40310}(1099,\cdot)\)
\(\chi_{40310}(1129,\cdot)\)
\(\chi_{40310}(1319,\cdot)\)
\(\chi_{40310}(1349,\cdot)\)
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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 ]
\((24187,19461,16821)\) → \((-1,e\left(\frac{9}{28}\right),e\left(\frac{61}{138}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(19, a) \) |
\(1\) | \(1\) | \(e\left(\frac{445}{1932}\right)\) | \(e\left(\frac{443}{966}\right)\) | \(e\left(\frac{445}{966}\right)\) | \(e\left(\frac{1217}{1932}\right)\) | \(e\left(\frac{278}{483}\right)\) | \(e\left(\frac{151}{276}\right)\) | \(e\left(\frac{1655}{1932}\right)\) | \(e\left(\frac{1331}{1932}\right)\) | \(e\left(\frac{139}{161}\right)\) | \(e\left(\frac{445}{644}\right)\) |
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sage:chi.jacobi_sum(n)
