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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15800, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,13,0,22]))
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pari:[g,chi] = znchar(Mod(2301,15800))
\(\chi_{15800}(101,\cdot)\)
\(\chi_{15800}(301,\cdot)\)
\(\chi_{15800}(2301,\cdot)\)
\(\chi_{15800}(2501,\cdot)\)
\(\chi_{15800}(3301,\cdot)\)
\(\chi_{15800}(3701,\cdot)\)
\(\chi_{15800}(5301,\cdot)\)
\(\chi_{15800}(6101,\cdot)\)
\(\chi_{15800}(6701,\cdot)\)
\(\chi_{15800}(7701,\cdot)\)
\(\chi_{15800}(9501,\cdot)\)
\(\chi_{15800}(11501,\cdot)\)
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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 ]
\((3951,7901,11377,12801)\) → \((1,-1,1,e\left(\frac{11}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 15800 }(2301, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(1\) | \(e\left(\frac{1}{26}\right)\) |
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sage:chi.jacobi_sum(n)
