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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(345, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,2]))
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pari:[g,chi] = znchar(Mod(28,345))
\(\chi_{345}(7,\cdot)\)
\(\chi_{345}(28,\cdot)\)
\(\chi_{345}(37,\cdot)\)
\(\chi_{345}(43,\cdot)\)
\(\chi_{345}(67,\cdot)\)
\(\chi_{345}(88,\cdot)\)
\(\chi_{345}(97,\cdot)\)
\(\chi_{345}(103,\cdot)\)
\(\chi_{345}(112,\cdot)\)
\(\chi_{345}(148,\cdot)\)
\(\chi_{345}(157,\cdot)\)
\(\chi_{345}(172,\cdot)\)
\(\chi_{345}(178,\cdot)\)
\(\chi_{345}(217,\cdot)\)
\(\chi_{345}(247,\cdot)\)
\(\chi_{345}(268,\cdot)\)
\(\chi_{345}(283,\cdot)\)
\(\chi_{345}(313,\cdot)\)
\(\chi_{345}(337,\cdot)\)
\(\chi_{345}(343,\cdot)\)
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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 ]
\((116,277,166)\) → \((1,-i,e\left(\frac{1}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 345 }(28, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) |
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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)
