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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(916, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,28]))
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pari:[g,chi] = znchar(Mod(485,916))
\(\chi_{916}(17,\cdot)\)
\(\chi_{916}(53,\cdot)\)
\(\chi_{916}(57,\cdot)\)
\(\chi_{916}(61,\cdot)\)
\(\chi_{916}(121,\cdot)\)
\(\chi_{916}(161,\cdot)\)
\(\chi_{916}(165,\cdot)\)
\(\chi_{916}(225,\cdot)\)
\(\chi_{916}(245,\cdot)\)
\(\chi_{916}(273,\cdot)\)
\(\chi_{916}(289,\cdot)\)
\(\chi_{916}(333,\cdot)\)
\(\chi_{916}(485,\cdot)\)
\(\chi_{916}(501,\cdot)\)
\(\chi_{916}(661,\cdot)\)
\(\chi_{916}(729,\cdot)\)
\(\chi_{916}(901,\cdot)\)
\(\chi_{916}(905,\cdot)\)
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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 ]
\((459,693)\) → \((1,e\left(\frac{14}{19}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 916 }(485, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) |
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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)
