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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([1449,1863,686]))
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pari:[g,chi] = znchar(Mod(73,40310))
\(\chi_{40310}(73,\cdot)\)
\(\chi_{40310}(363,\cdot)\)
\(\chi_{40310}(443,\cdot)\)
\(\chi_{40310}(467,\cdot)\)
\(\chi_{40310}(507,\cdot)\)
\(\chi_{40310}(657,\cdot)\)
\(\chi_{40310}(707,\cdot)\)
\(\chi_{40310}(717,\cdot)\)
\(\chi_{40310}(727,\cdot)\)
\(\chi_{40310}(793,\cdot)\)
\(\chi_{40310}(797,\cdot)\)
\(\chi_{40310}(943,\cdot)\)
\(\chi_{40310}(1013,\cdot)\)
\(\chi_{40310}(1023,\cdot)\)
\(\chi_{40310}(1083,\cdot)\)
\(\chi_{40310}(1087,\cdot)\)
\(\chi_{40310}(1273,\cdot)\)
\(\chi_{40310}(1307,\cdot)\)
\(\chi_{40310}(1323,\cdot)\)
\(\chi_{40310}(1377,\cdot)\)
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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 ]
\((24187,19461,16821)\) → \((-i,e\left(\frac{27}{28}\right),e\left(\frac{49}{138}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(73, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{304}{483}\right)\) | \(e\left(\frac{145}{1932}\right)\) | \(e\left(\frac{125}{483}\right)\) | \(e\left(\frac{179}{1932}\right)\) | \(e\left(\frac{641}{1932}\right)\) | \(e\left(\frac{137}{138}\right)\) | \(e\left(\frac{1619}{1932}\right)\) | \(e\left(\frac{1361}{1932}\right)\) | \(e\left(\frac{79}{644}\right)\) | \(e\left(\frac{143}{161}\right)\) |
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sage:chi.jacobi_sum(n)
