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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([1449,345,574]))
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pari:[g,chi] = znchar(Mod(3,40310))
\(\chi_{40310}(3,\cdot)\)
\(\chi_{40310}(243,\cdot)\)
\(\chi_{40310}(247,\cdot)\)
\(\chi_{40310}(253,\cdot)\)
\(\chi_{40310}(293,\cdot)\)
\(\chi_{40310}(387,\cdot)\)
\(\chi_{40310}(543,\cdot)\)
\(\chi_{40310}(577,\cdot)\)
\(\chi_{40310}(617,\cdot)\)
\(\chi_{40310}(823,\cdot)\)
\(\chi_{40310}(827,\cdot)\)
\(\chi_{40310}(907,\cdot)\)
\(\chi_{40310}(1063,\cdot)\)
\(\chi_{40310}(1133,\cdot)\)
\(\chi_{40310}(1197,\cdot)\)
\(\chi_{40310}(1353,\cdot)\)
\(\chi_{40310}(1407,\cdot)\)
\(\chi_{40310}(1547,\cdot)\)
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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 ]
\((24187,19461,16821)\) → \((-i,e\left(\frac{5}{28}\right),e\left(\frac{41}{138}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(3, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{313}{966}\right)\) | \(e\left(\frac{1445}{1932}\right)\) | \(e\left(\frac{313}{483}\right)\) | \(e\left(\frac{85}{1932}\right)\) | \(e\left(\frac{925}{1932}\right)\) | \(e\left(\frac{20}{69}\right)\) | \(e\left(\frac{445}{1932}\right)\) | \(e\left(\frac{139}{1932}\right)\) | \(e\left(\frac{543}{644}\right)\) | \(e\left(\frac{313}{322}\right)\) |
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sage:chi.jacobi_sum(n)
