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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(162, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([1]))
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pari:[g,chi] = znchar(Mod(83,162))
\(\chi_{162}(5,\cdot)\)
\(\chi_{162}(11,\cdot)\)
\(\chi_{162}(23,\cdot)\)
\(\chi_{162}(29,\cdot)\)
\(\chi_{162}(41,\cdot)\)
\(\chi_{162}(47,\cdot)\)
\(\chi_{162}(59,\cdot)\)
\(\chi_{162}(65,\cdot)\)
\(\chi_{162}(77,\cdot)\)
\(\chi_{162}(83,\cdot)\)
\(\chi_{162}(95,\cdot)\)
\(\chi_{162}(101,\cdot)\)
\(\chi_{162}(113,\cdot)\)
\(\chi_{162}(119,\cdot)\)
\(\chi_{162}(131,\cdot)\)
\(\chi_{162}(137,\cdot)\)
\(\chi_{162}(149,\cdot)\)
\(\chi_{162}(155,\cdot)\)
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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 ]
\(83\) → \(e\left(\frac{1}{54}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 162 }(83, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) |
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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)
