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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(555, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,14]))
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pari:[g,chi] = znchar(Mod(67,555))
\(\chi_{555}(28,\cdot)\)
\(\chi_{555}(58,\cdot)\)
\(\chi_{555}(67,\cdot)\)
\(\chi_{555}(178,\cdot)\)
\(\chi_{555}(247,\cdot)\)
\(\chi_{555}(262,\cdot)\)
\(\chi_{555}(337,\cdot)\)
\(\chi_{555}(358,\cdot)\)
\(\chi_{555}(373,\cdot)\)
\(\chi_{555}(448,\cdot)\)
\(\chi_{555}(472,\cdot)\)
\(\chi_{555}(502,\cdot)\)
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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 ]
\((371,112,76)\) → \((1,i,e\left(\frac{7}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 555 }(67, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) |
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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)
