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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(676, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([0,101]))
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pari:[g,chi] = znchar(Mod(45,676))
\(\chi_{676}(33,\cdot)\)
\(\chi_{676}(37,\cdot)\)
\(\chi_{676}(41,\cdot)\)
\(\chi_{676}(45,\cdot)\)
\(\chi_{676}(85,\cdot)\)
\(\chi_{676}(93,\cdot)\)
\(\chi_{676}(97,\cdot)\)
\(\chi_{676}(137,\cdot)\)
\(\chi_{676}(141,\cdot)\)
\(\chi_{676}(145,\cdot)\)
\(\chi_{676}(149,\cdot)\)
\(\chi_{676}(189,\cdot)\)
\(\chi_{676}(193,\cdot)\)
\(\chi_{676}(197,\cdot)\)
\(\chi_{676}(201,\cdot)\)
\(\chi_{676}(241,\cdot)\)
\(\chi_{676}(245,\cdot)\)
\(\chi_{676}(253,\cdot)\)
\(\chi_{676}(293,\cdot)\)
\(\chi_{676}(297,\cdot)\)
\(\chi_{676}(301,\cdot)\)
\(\chi_{676}(305,\cdot)\)
\(\chi_{676}(345,\cdot)\)
\(\chi_{676}(349,\cdot)\)
\(\chi_{676}(353,\cdot)\)
\(\chi_{676}(397,\cdot)\)
\(\chi_{676}(401,\cdot)\)
\(\chi_{676}(405,\cdot)\)
\(\chi_{676}(409,\cdot)\)
\(\chi_{676}(449,\cdot)\)
...
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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 ]
\((339,509)\) → \((1,e\left(\frac{101}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 676 }(45, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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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)
