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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,13]))
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pari:[g,chi] = znchar(Mod(43,87))
\(\chi_{87}(10,\cdot)\)
\(\chi_{87}(19,\cdot)\)
\(\chi_{87}(31,\cdot)\)
\(\chi_{87}(37,\cdot)\)
\(\chi_{87}(40,\cdot)\)
\(\chi_{87}(43,\cdot)\)
\(\chi_{87}(55,\cdot)\)
\(\chi_{87}(61,\cdot)\)
\(\chi_{87}(73,\cdot)\)
\(\chi_{87}(76,\cdot)\)
\(\chi_{87}(79,\cdot)\)
\(\chi_{87}(85,\cdot)\)
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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 ]
\((59,31)\) → \((1,e\left(\frac{13}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 87 }(43, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) |
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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)
