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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4600, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,33,30]))
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pari:[g,chi] = znchar(Mod(893,4600))
\(\chi_{4600}(157,\cdot)\)
\(\chi_{4600}(293,\cdot)\)
\(\chi_{4600}(493,\cdot)\)
\(\chi_{4600}(557,\cdot)\)
\(\chi_{4600}(757,\cdot)\)
\(\chi_{4600}(893,\cdot)\)
\(\chi_{4600}(957,\cdot)\)
\(\chi_{4600}(1157,\cdot)\)
\(\chi_{4600}(1293,\cdot)\)
\(\chi_{4600}(1493,\cdot)\)
\(\chi_{4600}(1693,\cdot)\)
\(\chi_{4600}(1893,\cdot)\)
\(\chi_{4600}(2357,\cdot)\)
\(\chi_{4600}(2757,\cdot)\)
\(\chi_{4600}(3093,\cdot)\)
\(\chi_{4600}(3493,\cdot)\)
\(\chi_{4600}(3557,\cdot)\)
\(\chi_{4600}(4157,\cdot)\)
\(\chi_{4600}(4293,\cdot)\)
\(\chi_{4600}(4357,\cdot)\)
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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 ]
\((1151,2301,2577,1201)\) → \((1,-1,-i,e\left(\frac{15}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 4600 }(893, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) |
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sage:chi.jacobi_sum(n)
