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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,13,13,18]))
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pari:[g,chi] = znchar(Mod(5591,8112))
\(\chi_{8112}(599,\cdot)\)
\(\chi_{8112}(1223,\cdot)\)
\(\chi_{8112}(1847,\cdot)\)
\(\chi_{8112}(2471,\cdot)\)
\(\chi_{8112}(3095,\cdot)\)
\(\chi_{8112}(4343,\cdot)\)
\(\chi_{8112}(4967,\cdot)\)
\(\chi_{8112}(5591,\cdot)\)
\(\chi_{8112}(6215,\cdot)\)
\(\chi_{8112}(6839,\cdot)\)
\(\chi_{8112}(7463,\cdot)\)
\(\chi_{8112}(8087,\cdot)\)
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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 ]
\((5071,6085,2705,3889)\) → \((-1,-1,-1,e\left(\frac{9}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(5591, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) |
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sage:chi.jacobi_sum(n)
