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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(398, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([10]))
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pari:[g,chi] = znchar(Mod(387,398))
\(\chi_{398}(61,\cdot)\)
\(\chi_{398}(63,\cdot)\)
\(\chi_{398}(103,\cdot)\)
\(\chi_{398}(121,\cdot)\)
\(\chi_{398}(125,\cdot)\)
\(\chi_{398}(139,\cdot)\)
\(\chi_{398}(217,\cdot)\)
\(\chi_{398}(261,\cdot)\)
\(\chi_{398}(313,\cdot)\)
\(\chi_{398}(387,\cdot)\)
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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 ]
\(3\) → \(e\left(\frac{5}{11}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 398 }(387, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(1\) | \(1\) |
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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)
