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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(242, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([14]))
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pari:[g,chi] = znchar(Mod(23,242))
\(\chi_{242}(23,\cdot)\)
\(\chi_{242}(45,\cdot)\)
\(\chi_{242}(67,\cdot)\)
\(\chi_{242}(89,\cdot)\)
\(\chi_{242}(111,\cdot)\)
\(\chi_{242}(133,\cdot)\)
\(\chi_{242}(155,\cdot)\)
\(\chi_{242}(177,\cdot)\)
\(\chi_{242}(199,\cdot)\)
\(\chi_{242}(221,\cdot)\)
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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 ]
\(123\) → \(e\left(\frac{7}{11}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 242 }(23, a) \) |
\(1\) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
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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)
