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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(46, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([13]))
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pari:[g,chi] = znchar(Mod(21,46))
\(\chi_{46}(5,\cdot)\)
\(\chi_{46}(7,\cdot)\)
\(\chi_{46}(11,\cdot)\)
\(\chi_{46}(15,\cdot)\)
\(\chi_{46}(17,\cdot)\)
\(\chi_{46}(19,\cdot)\)
\(\chi_{46}(21,\cdot)\)
\(\chi_{46}(33,\cdot)\)
\(\chi_{46}(37,\cdot)\)
\(\chi_{46}(43,\cdot)\)
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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 ]
\(5\) → \(e\left(\frac{13}{22}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 46 }(21, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) |
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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)
