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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,17,0]))
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pari:[g,chi] = znchar(Mod(397,825))
\(\chi_{825}(67,\cdot)\)
\(\chi_{825}(133,\cdot)\)
\(\chi_{825}(298,\cdot)\)
\(\chi_{825}(397,\cdot)\)
\(\chi_{825}(463,\cdot)\)
\(\chi_{825}(562,\cdot)\)
\(\chi_{825}(628,\cdot)\)
\(\chi_{825}(727,\cdot)\)
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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 ]
\((551,727,376)\) → \((1,e\left(\frac{17}{20}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 825 }(397, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) |
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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)
