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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(64))
M = H._module
chi = DirichletCharacter(H, M([32,51,0]))
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pari:[g,chi] = znchar(Mod(67,768))
\(\chi_{768}(19,\cdot)\)
\(\chi_{768}(43,\cdot)\)
\(\chi_{768}(67,\cdot)\)
\(\chi_{768}(91,\cdot)\)
\(\chi_{768}(115,\cdot)\)
\(\chi_{768}(139,\cdot)\)
\(\chi_{768}(163,\cdot)\)
\(\chi_{768}(187,\cdot)\)
\(\chi_{768}(211,\cdot)\)
\(\chi_{768}(235,\cdot)\)
\(\chi_{768}(259,\cdot)\)
\(\chi_{768}(283,\cdot)\)
\(\chi_{768}(307,\cdot)\)
\(\chi_{768}(331,\cdot)\)
\(\chi_{768}(355,\cdot)\)
\(\chi_{768}(379,\cdot)\)
\(\chi_{768}(403,\cdot)\)
\(\chi_{768}(427,\cdot)\)
\(\chi_{768}(451,\cdot)\)
\(\chi_{768}(475,\cdot)\)
\(\chi_{768}(499,\cdot)\)
\(\chi_{768}(523,\cdot)\)
\(\chi_{768}(547,\cdot)\)
\(\chi_{768}(571,\cdot)\)
\(\chi_{768}(595,\cdot)\)
\(\chi_{768}(619,\cdot)\)
\(\chi_{768}(643,\cdot)\)
\(\chi_{768}(667,\cdot)\)
\(\chi_{768}(691,\cdot)\)
\(\chi_{768}(715,\cdot)\)
...
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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 ]
\((511,517,257)\) → \((-1,e\left(\frac{51}{64}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 768 }(67, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
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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)
