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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([483,207,1120]))
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pari:[g,chi] = znchar(Mod(37,40310))
\(\chi_{40310}(37,\cdot)\)
\(\chi_{40310}(47,\cdot)\)
\(\chi_{40310}(193,\cdot)\)
\(\chi_{40310}(263,\cdot)\)
\(\chi_{40310}(287,\cdot)\)
\(\chi_{40310}(327,\cdot)\)
\(\chi_{40310}(483,\cdot)\)
\(\chi_{40310}(537,\cdot)\)
\(\chi_{40310}(553,\cdot)\)
\(\chi_{40310}(607,\cdot)\)
\(\chi_{40310}(623,\cdot)\)
\(\chi_{40310}(627,\cdot)\)
\(\chi_{40310}(677,\cdot)\)
\(\chi_{40310}(773,\cdot)\)
\(\chi_{40310}(843,\cdot)\)
\(\chi_{40310}(917,\cdot)\)
\(\chi_{40310}(1117,\cdot)\)
\(\chi_{40310}(1123,\cdot)\)
\(\chi_{40310}(1163,\cdot)\)
\(\chi_{40310}(1403,\cdot)\)
\(\chi_{40310}(1497,\cdot)\)
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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 ]
\((24187,19461,16821)\) → \((i,e\left(\frac{3}{28}\right),e\left(\frac{40}{69}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(37, a) \) |
\(1\) | \(1\) | \(e\left(\frac{26}{483}\right)\) | \(e\left(\frac{1007}{1932}\right)\) | \(e\left(\frac{52}{483}\right)\) | \(e\left(\frac{1423}{1932}\right)\) | \(e\left(\frac{1507}{1932}\right)\) | \(e\left(\frac{73}{138}\right)\) | \(e\left(\frac{1597}{1932}\right)\) | \(e\left(\frac{1111}{1932}\right)\) | \(e\left(\frac{351}{644}\right)\) | \(e\left(\frac{26}{161}\right)\) |
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sage:chi.jacobi_sum(n)
