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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(663, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,32,33]))
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pari:[g,chi] = znchar(Mod(568,663))
\(\chi_{663}(22,\cdot)\)
\(\chi_{663}(61,\cdot)\)
\(\chi_{663}(133,\cdot)\)
\(\chi_{663}(139,\cdot)\)
\(\chi_{663}(211,\cdot)\)
\(\chi_{663}(250,\cdot)\)
\(\chi_{663}(295,\cdot)\)
\(\chi_{663}(328,\cdot)\)
\(\chi_{663}(334,\cdot)\)
\(\chi_{663}(367,\cdot)\)
\(\chi_{663}(445,\cdot)\)
\(\chi_{663}(490,\cdot)\)
\(\chi_{663}(568,\cdot)\)
\(\chi_{663}(601,\cdot)\)
\(\chi_{663}(607,\cdot)\)
\(\chi_{663}(640,\cdot)\)
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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 ]
\((443,613,547)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{11}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(19\) |
| \( \chi_{ 663 }(568, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{24}\right)\) |
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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)
