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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(82, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([7]))
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pari:[g,chi] = znchar(Mod(29,82))
\(\chi_{82}(7,\cdot)\)
\(\chi_{82}(11,\cdot)\)
\(\chi_{82}(13,\cdot)\)
\(\chi_{82}(15,\cdot)\)
\(\chi_{82}(17,\cdot)\)
\(\chi_{82}(19,\cdot)\)
\(\chi_{82}(29,\cdot)\)
\(\chi_{82}(35,\cdot)\)
\(\chi_{82}(47,\cdot)\)
\(\chi_{82}(53,\cdot)\)
\(\chi_{82}(63,\cdot)\)
\(\chi_{82}(65,\cdot)\)
\(\chi_{82}(67,\cdot)\)
\(\chi_{82}(69,\cdot)\)
\(\chi_{82}(71,\cdot)\)
\(\chi_{82}(75,\cdot)\)
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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 ]
\(47\) → \(e\left(\frac{7}{40}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 82 }(29, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(i\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) |
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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)
