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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(259, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,17]))
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pari:[g,chi] = znchar(Mod(92,259))
\(\chi_{259}(15,\cdot)\)
\(\chi_{259}(22,\cdot)\)
\(\chi_{259}(50,\cdot)\)
\(\chi_{259}(57,\cdot)\)
\(\chi_{259}(92,\cdot)\)
\(\chi_{259}(106,\cdot)\)
\(\chi_{259}(113,\cdot)\)
\(\chi_{259}(183,\cdot)\)
\(\chi_{259}(190,\cdot)\)
\(\chi_{259}(204,\cdot)\)
\(\chi_{259}(239,\cdot)\)
\(\chi_{259}(246,\cdot)\)
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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 ]
\((38,113)\) → \((1,e\left(\frac{17}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 259 }(92, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(-i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) |
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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)
