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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(736, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,6]))
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pari:[g,chi] = znchar(Mod(721,736))
\(\chi_{736}(49,\cdot)\)
\(\chi_{736}(81,\cdot)\)
\(\chi_{736}(177,\cdot)\)
\(\chi_{736}(209,\cdot)\)
\(\chi_{736}(305,\cdot)\)
\(\chi_{736}(561,\cdot)\)
\(\chi_{736}(593,\cdot)\)
\(\chi_{736}(625,\cdot)\)
\(\chi_{736}(657,\cdot)\)
\(\chi_{736}(721,\cdot)\)
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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 ]
\((415,645,97)\) → \((1,-1,e\left(\frac{3}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 736 }(721, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) |
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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)
