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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(828, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,1]))
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pari:[g,chi] = znchar(Mod(143,828))
\(\chi_{828}(107,\cdot)\)
\(\chi_{828}(143,\cdot)\)
\(\chi_{828}(251,\cdot)\)
\(\chi_{828}(287,\cdot)\)
\(\chi_{828}(359,\cdot)\)
\(\chi_{828}(431,\cdot)\)
\(\chi_{828}(467,\cdot)\)
\(\chi_{828}(503,\cdot)\)
\(\chi_{828}(539,\cdot)\)
\(\chi_{828}(755,\cdot)\)
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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 ]
\((415,461,649)\) → \((-1,-1,e\left(\frac{1}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 828 }(143, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) |
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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)
