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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([27,22]))
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pari:[g,chi] = znchar(Mod(243,475))
\(\chi_{475}(32,\cdot)\)
\(\chi_{475}(143,\cdot)\)
\(\chi_{475}(193,\cdot)\)
\(\chi_{475}(243,\cdot)\)
\(\chi_{475}(257,\cdot)\)
\(\chi_{475}(268,\cdot)\)
\(\chi_{475}(307,\cdot)\)
\(\chi_{475}(318,\cdot)\)
\(\chi_{475}(357,\cdot)\)
\(\chi_{475}(382,\cdot)\)
\(\chi_{475}(393,\cdot)\)
\(\chi_{475}(432,\cdot)\)
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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 ]
\((77,401)\) → \((-i,e\left(\frac{11}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 475 }(243, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{36}\right)\) |
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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)
