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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8820, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,14,0,25]))
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pari:[g,chi] = znchar(Mod(1291,8820))
\(\chi_{8820}(691,\cdot)\)
\(\chi_{8820}(1291,\cdot)\)
\(\chi_{8820}(1951,\cdot)\)
\(\chi_{8820}(2551,\cdot)\)
\(\chi_{8820}(3211,\cdot)\)
\(\chi_{8820}(3811,\cdot)\)
\(\chi_{8820}(4471,\cdot)\)
\(\chi_{8820}(5071,\cdot)\)
\(\chi_{8820}(5731,\cdot)\)
\(\chi_{8820}(6331,\cdot)\)
\(\chi_{8820}(6991,\cdot)\)
\(\chi_{8820}(7591,\cdot)\)
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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 ]
\((4411,7841,7057,1081)\) → \((-1,e\left(\frac{1}{3}\right),1,e\left(\frac{25}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8820 }(1291, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) |
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sage:chi.jacobi_sum(n)
