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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([0,30,15,10,18]))
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pari:[g,chi] = znchar(Mod(437,4620))
\(\chi_{4620}(17,\cdot)\)
\(\chi_{4620}(173,\cdot)\)
\(\chi_{4620}(437,\cdot)\)
\(\chi_{4620}(677,\cdot)\)
\(\chi_{4620}(1097,\cdot)\)
\(\chi_{4620}(1613,\cdot)\)
\(\chi_{4620}(2273,\cdot)\)
\(\chi_{4620}(2537,\cdot)\)
\(\chi_{4620}(2873,\cdot)\)
\(\chi_{4620}(3197,\cdot)\)
\(\chi_{4620}(3533,\cdot)\)
\(\chi_{4620}(3713,\cdot)\)
\(\chi_{4620}(3797,\cdot)\)
\(\chi_{4620}(4133,\cdot)\)
\(\chi_{4620}(4373,\cdot)\)
\(\chi_{4620}(4457,\cdot)\)
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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 ]
\((2311,1541,3697,661,2521)\) → \((1,-1,i,e\left(\frac{1}{6}\right),e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4620 }(437, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(i\) | \(e\left(\frac{59}{60}\right)\) |
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sage:chi.jacobi_sum(n)
