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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([63,39,70]))
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pari:[g,chi] = znchar(Mod(43,40310))
\(\chi_{40310}(43,\cdot)\)
\(\chi_{40310}(97,\cdot)\)
\(\chi_{40310}(1487,\cdot)\)
\(\chi_{40310}(3433,\cdot)\)
\(\chi_{40310}(7827,\cdot)\)
\(\chi_{40310}(8383,\cdot)\)
\(\chi_{40310}(8437,\cdot)\)
\(\chi_{40310}(8993,\cdot)\)
\(\chi_{40310}(13387,\cdot)\)
\(\chi_{40310}(15333,\cdot)\)
\(\chi_{40310}(16723,\cdot)\)
\(\chi_{40310}(16777,\cdot)\)
\(\chi_{40310}(18947,\cdot)\)
\(\chi_{40310}(20337,\cdot)\)
\(\chi_{40310}(21503,\cdot)\)
\(\chi_{40310}(22283,\cdot)\)
\(\chi_{40310}(27287,\cdot)\)
\(\chi_{40310}(27843,\cdot)\)
\(\chi_{40310}(29287,\cdot)\)
\(\chi_{40310}(29843,\cdot)\)
\(\chi_{40310}(34847,\cdot)\)
\(\chi_{40310}(35627,\cdot)\)
\(\chi_{40310}(36793,\cdot)\)
\(\chi_{40310}(38183,\cdot)\)
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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 ]
\((24187,19461,16821)\) → \((-i,e\left(\frac{13}{28}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(43, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) |
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sage:chi.jacobi_sum(n)
