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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8712, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,0,18]))
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pari:[g,chi] = znchar(Mod(5941,8712))
\(\chi_{8712}(397,\cdot)\)
\(\chi_{8712}(1189,\cdot)\)
\(\chi_{8712}(1981,\cdot)\)
\(\chi_{8712}(2773,\cdot)\)
\(\chi_{8712}(3565,\cdot)\)
\(\chi_{8712}(5149,\cdot)\)
\(\chi_{8712}(5941,\cdot)\)
\(\chi_{8712}(6733,\cdot)\)
\(\chi_{8712}(7525,\cdot)\)
\(\chi_{8712}(8317,\cdot)\)
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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 ]
\((6535,4357,1937,5689)\) → \((1,-1,1,e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8712 }(5941, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) |
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sage:chi.jacobi_sum(n)
