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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(656, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,0,1]))
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pari:[g,chi] = znchar(Mod(47,656))
\(\chi_{656}(15,\cdot)\)
\(\chi_{656}(47,\cdot)\)
\(\chi_{656}(63,\cdot)\)
\(\chi_{656}(95,\cdot)\)
\(\chi_{656}(111,\cdot)\)
\(\chi_{656}(175,\cdot)\)
\(\chi_{656}(239,\cdot)\)
\(\chi_{656}(335,\cdot)\)
\(\chi_{656}(399,\cdot)\)
\(\chi_{656}(463,\cdot)\)
\(\chi_{656}(479,\cdot)\)
\(\chi_{656}(511,\cdot)\)
\(\chi_{656}(527,\cdot)\)
\(\chi_{656}(559,\cdot)\)
\(\chi_{656}(591,\cdot)\)
\(\chi_{656}(639,\cdot)\)
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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 ]
\((575,165,129)\) → \((-1,1,e\left(\frac{1}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 656 }(47, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(-i\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) |
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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)
