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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,1]))
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pari:[g,chi] = znchar(Mod(224,425))
\(\chi_{425}(24,\cdot)\)
\(\chi_{425}(74,\cdot)\)
\(\chi_{425}(99,\cdot)\)
\(\chi_{425}(124,\cdot)\)
\(\chi_{425}(199,\cdot)\)
\(\chi_{425}(224,\cdot)\)
\(\chi_{425}(249,\cdot)\)
\(\chi_{425}(299,\cdot)\)
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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 ]
\((52,326)\) → \((-1,e\left(\frac{1}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 425 }(224, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(-i\) |
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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)
