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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(558, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([10,14]))
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pari:[g,chi] = znchar(Mod(103,558))
\(\chi_{558}(7,\cdot)\)
\(\chi_{558}(49,\cdot)\)
\(\chi_{558}(103,\cdot)\)
\(\chi_{558}(169,\cdot)\)
\(\chi_{558}(175,\cdot)\)
\(\chi_{558}(205,\cdot)\)
\(\chi_{558}(319,\cdot)\)
\(\chi_{558}(493,\cdot)\)
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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 ]
\((497,127)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(35\) |
| \( \chi_{ 558 }(103, a) \) |
\(1\) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) |
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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)
