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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4600, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,0,44,100]))
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pari:[g,chi] = znchar(Mod(81,4600))
\(\chi_{4600}(41,\cdot)\)
\(\chi_{4600}(81,\cdot)\)
\(\chi_{4600}(121,\cdot)\)
\(\chi_{4600}(361,\cdot)\)
\(\chi_{4600}(441,\cdot)\)
\(\chi_{4600}(561,\cdot)\)
\(\chi_{4600}(721,\cdot)\)
\(\chi_{4600}(761,\cdot)\)
\(\chi_{4600}(841,\cdot)\)
\(\chi_{4600}(961,\cdot)\)
\(\chi_{4600}(1041,\cdot)\)
\(\chi_{4600}(1281,\cdot)\)
\(\chi_{4600}(1361,\cdot)\)
\(\chi_{4600}(1481,\cdot)\)
\(\chi_{4600}(1521,\cdot)\)
\(\chi_{4600}(1641,\cdot)\)
\(\chi_{4600}(1681,\cdot)\)
\(\chi_{4600}(1761,\cdot)\)
\(\chi_{4600}(1881,\cdot)\)
\(\chi_{4600}(1921,\cdot)\)
\(\chi_{4600}(1961,\cdot)\)
\(\chi_{4600}(2281,\cdot)\)
\(\chi_{4600}(2441,\cdot)\)
\(\chi_{4600}(2561,\cdot)\)
\(\chi_{4600}(2681,\cdot)\)
\(\chi_{4600}(2841,\cdot)\)
\(\chi_{4600}(2881,\cdot)\)
\(\chi_{4600}(3121,\cdot)\)
\(\chi_{4600}(3321,\cdot)\)
\(\chi_{4600}(3361,\cdot)\)
...
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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 ]
\((1151,2301,2577,1201)\) → \((1,1,e\left(\frac{2}{5}\right),e\left(\frac{10}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 4600 }(81, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) |
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sage:chi.jacobi_sum(n)
