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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8004, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,14,0,11]))
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pari:[g,chi] = znchar(Mod(47,8004))
\(\chi_{8004}(47,\cdot)\)
\(\chi_{8004}(599,\cdot)\)
\(\chi_{8004}(1703,\cdot)\)
\(\chi_{8004}(2531,\cdot)\)
\(\chi_{8004}(3635,\cdot)\)
\(\chi_{8004}(4187,\cdot)\)
\(\chi_{8004}(4463,\cdot)\)
\(\chi_{8004}(5015,\cdot)\)
\(\chi_{8004}(5843,\cdot)\)
\(\chi_{8004}(6395,\cdot)\)
\(\chi_{8004}(7223,\cdot)\)
\(\chi_{8004}(7775,\cdot)\)
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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 ]
\((4003,2669,3133,553)\) → \((-1,-1,1,e\left(\frac{11}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(47, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(-i\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) |
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sage:chi.jacobi_sum(n)
