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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(777, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,0,17]))
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pari:[g,chi] = znchar(Mod(610,777))
\(\chi_{777}(22,\cdot)\)
\(\chi_{777}(106,\cdot)\)
\(\chi_{777}(190,\cdot)\)
\(\chi_{777}(274,\cdot)\)
\(\chi_{777}(316,\cdot)\)
\(\chi_{777}(442,\cdot)\)
\(\chi_{777}(463,\cdot)\)
\(\chi_{777}(505,\cdot)\)
\(\chi_{777}(568,\cdot)\)
\(\chi_{777}(610,\cdot)\)
\(\chi_{777}(631,\cdot)\)
\(\chi_{777}(757,\cdot)\)
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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 ]
\((260,556,631)\) → \((1,1,e\left(\frac{17}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 777 }(610, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) |
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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)
