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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(680, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,0,0,9]))
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pari:[g,chi] = znchar(Mod(31,680))
\(\chi_{680}(31,\cdot)\)
\(\chi_{680}(71,\cdot)\)
\(\chi_{680}(231,\cdot)\)
\(\chi_{680}(311,\cdot)\)
\(\chi_{680}(351,\cdot)\)
\(\chi_{680}(431,\cdot)\)
\(\chi_{680}(471,\cdot)\)
\(\chi_{680}(551,\cdot)\)
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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 ]
\((511,341,137,241)\) → \((-1,1,1,e\left(\frac{9}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 680 }(31, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
