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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(243, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([52]))
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pari:[g,chi] = znchar(Mod(19,243))
\(\chi_{243}(10,\cdot)\)
\(\chi_{243}(19,\cdot)\)
\(\chi_{243}(37,\cdot)\)
\(\chi_{243}(46,\cdot)\)
\(\chi_{243}(64,\cdot)\)
\(\chi_{243}(73,\cdot)\)
\(\chi_{243}(91,\cdot)\)
\(\chi_{243}(100,\cdot)\)
\(\chi_{243}(118,\cdot)\)
\(\chi_{243}(127,\cdot)\)
\(\chi_{243}(145,\cdot)\)
\(\chi_{243}(154,\cdot)\)
\(\chi_{243}(172,\cdot)\)
\(\chi_{243}(181,\cdot)\)
\(\chi_{243}(199,\cdot)\)
\(\chi_{243}(208,\cdot)\)
\(\chi_{243}(226,\cdot)\)
\(\chi_{243}(235,\cdot)\)
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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 ]
\(2\) → \(e\left(\frac{26}{27}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 243 }(19, a) \) |
\(1\) | \(1\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) |
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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)
