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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(115, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,20]))
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pari:[g,chi] = znchar(Mod(81,115))
\(\chi_{115}(6,\cdot)\)
\(\chi_{115}(16,\cdot)\)
\(\chi_{115}(26,\cdot)\)
\(\chi_{115}(31,\cdot)\)
\(\chi_{115}(36,\cdot)\)
\(\chi_{115}(41,\cdot)\)
\(\chi_{115}(71,\cdot)\)
\(\chi_{115}(81,\cdot)\)
\(\chi_{115}(96,\cdot)\)
\(\chi_{115}(101,\cdot)\)
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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 ]
\((47,51)\) → \((1,e\left(\frac{10}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 115 }(81, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) |
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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)
