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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,13,0]))
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pari:[g,chi] = znchar(Mod(79,768))
\(\chi_{768}(79,\cdot)\)
\(\chi_{768}(175,\cdot)\)
\(\chi_{768}(271,\cdot)\)
\(\chi_{768}(367,\cdot)\)
\(\chi_{768}(463,\cdot)\)
\(\chi_{768}(559,\cdot)\)
\(\chi_{768}(655,\cdot)\)
\(\chi_{768}(751,\cdot)\)
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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 ]
\((511,517,257)\) → \((-1,e\left(\frac{13}{16}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 768 }(79, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(-i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(1\) |
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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)
