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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8820, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([0,28,21,6]))
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pari:[g,chi] = znchar(Mod(517,8820))
\(\chi_{8820}(13,\cdot)\)
\(\chi_{8820}(517,\cdot)\)
\(\chi_{8820}(853,\cdot)\)
\(\chi_{8820}(1357,\cdot)\)
\(\chi_{8820}(1777,\cdot)\)
\(\chi_{8820}(2113,\cdot)\)
\(\chi_{8820}(2533,\cdot)\)
\(\chi_{8820}(2617,\cdot)\)
\(\chi_{8820}(3373,\cdot)\)
\(\chi_{8820}(3793,\cdot)\)
\(\chi_{8820}(3877,\cdot)\)
\(\chi_{8820}(4297,\cdot)\)
\(\chi_{8820}(4633,\cdot)\)
\(\chi_{8820}(5053,\cdot)\)
\(\chi_{8820}(5137,\cdot)\)
\(\chi_{8820}(5557,\cdot)\)
\(\chi_{8820}(5893,\cdot)\)
\(\chi_{8820}(6313,\cdot)\)
\(\chi_{8820}(6397,\cdot)\)
\(\chi_{8820}(6817,\cdot)\)
\(\chi_{8820}(7573,\cdot)\)
\(\chi_{8820}(7657,\cdot)\)
\(\chi_{8820}(8077,\cdot)\)
\(\chi_{8820}(8413,\cdot)\)
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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 ]
\((4411,7841,7057,1081)\) → \((1,e\left(\frac{1}{3}\right),i,e\left(\frac{1}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8820 }(517, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(1\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{43}{84}\right)\) |
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sage:chi.jacobi_sum(n)
