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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(213, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([0,12]))
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pari:[g,chi] = znchar(Mod(4,213))
\(\chi_{213}(4,\cdot)\)
\(\chi_{213}(10,\cdot)\)
\(\chi_{213}(16,\cdot)\)
\(\chi_{213}(19,\cdot)\)
\(\chi_{213}(40,\cdot)\)
\(\chi_{213}(43,\cdot)\)
\(\chi_{213}(49,\cdot)\)
\(\chi_{213}(58,\cdot)\)
\(\chi_{213}(64,\cdot)\)
\(\chi_{213}(73,\cdot)\)
\(\chi_{213}(79,\cdot)\)
\(\chi_{213}(100,\cdot)\)
\(\chi_{213}(109,\cdot)\)
\(\chi_{213}(121,\cdot)\)
\(\chi_{213}(145,\cdot)\)
\(\chi_{213}(148,\cdot)\)
\(\chi_{213}(151,\cdot)\)
\(\chi_{213}(154,\cdot)\)
\(\chi_{213}(157,\cdot)\)
\(\chi_{213}(160,\cdot)\)
\(\chi_{213}(166,\cdot)\)
\(\chi_{213}(169,\cdot)\)
\(\chi_{213}(178,\cdot)\)
\(\chi_{213}(202,\cdot)\)
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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 ]
\((143,7)\) → \((1,e\left(\frac{6}{35}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 213 }(4, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{35}\right)\) |
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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)
