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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([0,18,35]))
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pari:[g,chi] = znchar(Mod(19,555))
\(\chi_{555}(19,\cdot)\)
\(\chi_{555}(79,\cdot)\)
\(\chi_{555}(94,\cdot)\)
\(\chi_{555}(109,\cdot)\)
\(\chi_{555}(124,\cdot)\)
\(\chi_{555}(244,\cdot)\)
\(\chi_{555}(274,\cdot)\)
\(\chi_{555}(394,\cdot)\)
\(\chi_{555}(409,\cdot)\)
\(\chi_{555}(424,\cdot)\)
\(\chi_{555}(439,\cdot)\)
\(\chi_{555}(499,\cdot)\)
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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 ]
\((371,112,76)\) → \((1,-1,e\left(\frac{35}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 555 }(19, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) |
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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)
