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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([7,5]))
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pari:[g,chi] = znchar(Mod(125,261))
\(\chi_{261}(35,\cdot)\)
\(\chi_{261}(62,\cdot)\)
\(\chi_{261}(71,\cdot)\)
\(\chi_{261}(80,\cdot)\)
\(\chi_{261}(125,\cdot)\)
\(\chi_{261}(179,\cdot)\)
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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 ]
\((146,118)\) → \((-1,e\left(\frac{5}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 261 }(125, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) |
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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)
