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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(158, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([32]))
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pari:[g,chi] = znchar(Mod(19,158))
\(\chi_{158}(5,\cdot)\)
\(\chi_{158}(9,\cdot)\)
\(\chi_{158}(11,\cdot)\)
\(\chi_{158}(13,\cdot)\)
\(\chi_{158}(19,\cdot)\)
\(\chi_{158}(25,\cdot)\)
\(\chi_{158}(31,\cdot)\)
\(\chi_{158}(45,\cdot)\)
\(\chi_{158}(49,\cdot)\)
\(\chi_{158}(51,\cdot)\)
\(\chi_{158}(73,\cdot)\)
\(\chi_{158}(81,\cdot)\)
\(\chi_{158}(83,\cdot)\)
\(\chi_{158}(95,\cdot)\)
\(\chi_{158}(99,\cdot)\)
\(\chi_{158}(105,\cdot)\)
\(\chi_{158}(111,\cdot)\)
\(\chi_{158}(115,\cdot)\)
\(\chi_{158}(119,\cdot)\)
\(\chi_{158}(121,\cdot)\)
\(\chi_{158}(123,\cdot)\)
\(\chi_{158}(129,\cdot)\)
\(\chi_{158}(151,\cdot)\)
\(\chi_{158}(155,\cdot)\)
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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 ]
\(3\) → \(e\left(\frac{16}{39}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 158 }(19, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{2}{13}\right)\) |
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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)
