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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,22,14]))
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pari:[g,chi] = znchar(Mod(293,4830))
\(\chi_{4830}(83,\cdot)\)
\(\chi_{4830}(293,\cdot)\)
\(\chi_{4830}(503,\cdot)\)
\(\chi_{4830}(797,\cdot)\)
\(\chi_{4830}(1217,\cdot)\)
\(\chi_{4830}(1763,\cdot)\)
\(\chi_{4830}(1847,\cdot)\)
\(\chi_{4830}(2057,\cdot)\)
\(\chi_{4830}(2183,\cdot)\)
\(\chi_{4830}(2687,\cdot)\)
\(\chi_{4830}(2813,\cdot)\)
\(\chi_{4830}(3023,\cdot)\)
\(\chi_{4830}(3317,\cdot)\)
\(\chi_{4830}(3653,\cdot)\)
\(\chi_{4830}(3737,\cdot)\)
\(\chi_{4830}(3947,\cdot)\)
\(\chi_{4830}(4157,\cdot)\)
\(\chi_{4830}(4283,\cdot)\)
\(\chi_{4830}(4367,\cdot)\)
\(\chi_{4830}(4703,\cdot)\)
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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 ]
\((3221,967,2761,1891)\) → \((-1,-i,-1,e\left(\frac{7}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4830 }(293, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
