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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8007, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([0,39,46]))
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pari:[g,chi] = znchar(Mod(4288,8007))
\(\chi_{8007}(4,\cdot)\)
\(\chi_{8007}(64,\cdot)\)
\(\chi_{8007}(268,\cdot)\)
\(\chi_{8007}(370,\cdot)\)
\(\chi_{8007}(718,\cdot)\)
\(\chi_{8007}(769,\cdot)\)
\(\chi_{8007}(1024,\cdot)\)
\(\chi_{8007}(1126,\cdot)\)
\(\chi_{8007}(1942,\cdot)\)
\(\chi_{8007}(2002,\cdot)\)
\(\chi_{8007}(2359,\cdot)\)
\(\chi_{8007}(3073,\cdot)\)
\(\chi_{8007}(3124,\cdot)\)
\(\chi_{8007}(3379,\cdot)\)
\(\chi_{8007}(3481,\cdot)\)
\(\chi_{8007}(4288,\cdot)\)
\(\chi_{8007}(4297,\cdot)\)
\(\chi_{8007}(4696,\cdot)\)
\(\chi_{8007}(5716,\cdot)\)
\(\chi_{8007}(5920,\cdot)\)
\(\chi_{8007}(6022,\cdot)\)
\(\chi_{8007}(6643,\cdot)\)
\(\chi_{8007}(7051,\cdot)\)
\(\chi_{8007}(7654,\cdot)\)
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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 ]
\((5339,1414,7855)\) → \((1,-i,e\left(\frac{23}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 8007 }(4288, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(1\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{12}{13}\right)\) |
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sage:chi.jacobi_sum(n)
