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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(775, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,14]))
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pari:[g,chi] = znchar(Mod(599,775))
\(\chi_{775}(49,\cdot)\)
\(\chi_{775}(174,\cdot)\)
\(\chi_{775}(224,\cdot)\)
\(\chi_{775}(299,\cdot)\)
\(\chi_{775}(324,\cdot)\)
\(\chi_{775}(474,\cdot)\)
\(\chi_{775}(524,\cdot)\)
\(\chi_{775}(599,\cdot)\)
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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 ]
\((652,251)\) → \((-1,e\left(\frac{7}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 775 }(599, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) |
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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)
