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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,1]))
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pari:[g,chi] = znchar(Mod(395,441))
\(\chi_{441}(17,\cdot)\)
\(\chi_{441}(26,\cdot)\)
\(\chi_{441}(89,\cdot)\)
\(\chi_{441}(143,\cdot)\)
\(\chi_{441}(152,\cdot)\)
\(\chi_{441}(206,\cdot)\)
\(\chi_{441}(269,\cdot)\)
\(\chi_{441}(278,\cdot)\)
\(\chi_{441}(332,\cdot)\)
\(\chi_{441}(341,\cdot)\)
\(\chi_{441}(395,\cdot)\)
\(\chi_{441}(404,\cdot)\)
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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 ]
\((344,199)\) → \((-1,e\left(\frac{1}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 441 }(395, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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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)
