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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([37]))
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pari:[g,chi] = znchar(Mod(89,243))
\(\chi_{243}(8,\cdot)\)
\(\chi_{243}(17,\cdot)\)
\(\chi_{243}(35,\cdot)\)
\(\chi_{243}(44,\cdot)\)
\(\chi_{243}(62,\cdot)\)
\(\chi_{243}(71,\cdot)\)
\(\chi_{243}(89,\cdot)\)
\(\chi_{243}(98,\cdot)\)
\(\chi_{243}(116,\cdot)\)
\(\chi_{243}(125,\cdot)\)
\(\chi_{243}(143,\cdot)\)
\(\chi_{243}(152,\cdot)\)
\(\chi_{243}(170,\cdot)\)
\(\chi_{243}(179,\cdot)\)
\(\chi_{243}(197,\cdot)\)
\(\chi_{243}(206,\cdot)\)
\(\chi_{243}(224,\cdot)\)
\(\chi_{243}(233,\cdot)\)
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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 ]
\(2\) → \(e\left(\frac{37}{54}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 243 }(89, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) |
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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)
