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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4620, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,15,20,27]))
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pari:[g,chi] = znchar(Mod(1579,4620))
\(\chi_{4620}(79,\cdot)\)
\(\chi_{4620}(739,\cdot)\)
\(\chi_{4620}(919,\cdot)\)
\(\chi_{4620}(1339,\cdot)\)
\(\chi_{4620}(1579,\cdot)\)
\(\chi_{4620}(1999,\cdot)\)
\(\chi_{4620}(3439,\cdot)\)
\(\chi_{4620}(4099,\cdot)\)
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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 ]
\((2311,1541,3697,661,2521)\) → \((-1,1,-1,e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4620 }(1579, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-1\) | \(e\left(\frac{8}{15}\right)\) |
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sage:chi.jacobi_sum(n)
