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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(512, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,9]))
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pari:[g,chi] = znchar(Mod(15,512))
\(\chi_{512}(15,\cdot)\)
\(\chi_{512}(47,\cdot)\)
\(\chi_{512}(79,\cdot)\)
\(\chi_{512}(111,\cdot)\)
\(\chi_{512}(143,\cdot)\)
\(\chi_{512}(175,\cdot)\)
\(\chi_{512}(207,\cdot)\)
\(\chi_{512}(239,\cdot)\)
\(\chi_{512}(271,\cdot)\)
\(\chi_{512}(303,\cdot)\)
\(\chi_{512}(335,\cdot)\)
\(\chi_{512}(367,\cdot)\)
\(\chi_{512}(399,\cdot)\)
\(\chi_{512}(431,\cdot)\)
\(\chi_{512}(463,\cdot)\)
\(\chi_{512}(495,\cdot)\)
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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 ]
\((511,5)\) → \((-1,e\left(\frac{9}{32}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 512 }(15, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) |
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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)
