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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4620, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,45,10,18]))
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pari:[g,chi] = znchar(Mod(283,4620))
\(\chi_{4620}(283,\cdot)\)
\(\chi_{4620}(523,\cdot)\)
\(\chi_{4620}(607,\cdot)\)
\(\chi_{4620}(787,\cdot)\)
\(\chi_{4620}(943,\cdot)\)
\(\chi_{4620}(1207,\cdot)\)
\(\chi_{4620}(1447,\cdot)\)
\(\chi_{4620}(1867,\cdot)\)
\(\chi_{4620}(2383,\cdot)\)
\(\chi_{4620}(3043,\cdot)\)
\(\chi_{4620}(3307,\cdot)\)
\(\chi_{4620}(3643,\cdot)\)
\(\chi_{4620}(3967,\cdot)\)
\(\chi_{4620}(4303,\cdot)\)
\(\chi_{4620}(4483,\cdot)\)
\(\chi_{4620}(4567,\cdot)\)
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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 ]
\((2311,1541,3697,661,2521)\) → \((-1,1,-i,e\left(\frac{1}{6}\right),e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4620 }(283, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) | \(e\left(\frac{29}{60}\right)\) |
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sage:chi.jacobi_sum(n)
