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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(110, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([5,16]))
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pari:[g,chi] = znchar(Mod(47,110))
\(\chi_{110}(3,\cdot)\)
\(\chi_{110}(27,\cdot)\)
\(\chi_{110}(37,\cdot)\)
\(\chi_{110}(47,\cdot)\)
\(\chi_{110}(53,\cdot)\)
\(\chi_{110}(93,\cdot)\)
\(\chi_{110}(97,\cdot)\)
\(\chi_{110}(103,\cdot)\)
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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 ]
\((67,101)\) → \((i,e\left(\frac{4}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 110 }(47, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
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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)
