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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,21]))
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pari:[g,chi] = znchar(Mod(83,322))
\(\chi_{322}(83,\cdot)\)
\(\chi_{322}(97,\cdot)\)
\(\chi_{322}(111,\cdot)\)
\(\chi_{322}(125,\cdot)\)
\(\chi_{322}(153,\cdot)\)
\(\chi_{322}(181,\cdot)\)
\(\chi_{322}(195,\cdot)\)
\(\chi_{322}(237,\cdot)\)
\(\chi_{322}(251,\cdot)\)
\(\chi_{322}(293,\cdot)\)
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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 ]
\((185,281)\) → \((-1,e\left(\frac{21}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 322 }(83, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) |
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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)
