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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([44]))
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pari:[g,chi] = znchar(Mod(67,162))
\(\chi_{162}(7,\cdot)\)
\(\chi_{162}(13,\cdot)\)
\(\chi_{162}(25,\cdot)\)
\(\chi_{162}(31,\cdot)\)
\(\chi_{162}(43,\cdot)\)
\(\chi_{162}(49,\cdot)\)
\(\chi_{162}(61,\cdot)\)
\(\chi_{162}(67,\cdot)\)
\(\chi_{162}(79,\cdot)\)
\(\chi_{162}(85,\cdot)\)
\(\chi_{162}(97,\cdot)\)
\(\chi_{162}(103,\cdot)\)
\(\chi_{162}(115,\cdot)\)
\(\chi_{162}(121,\cdot)\)
\(\chi_{162}(133,\cdot)\)
\(\chi_{162}(139,\cdot)\)
\(\chi_{162}(151,\cdot)\)
\(\chi_{162}(157,\cdot)\)
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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 ]
\(83\) → \(e\left(\frac{22}{27}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 162 }(67, a) \) |
\(1\) | \(1\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) |
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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)
