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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(875, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([27,40]))
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pari:[g,chi] = znchar(Mod(207,875))
\(\chi_{875}(18,\cdot)\)
\(\chi_{875}(32,\cdot)\)
\(\chi_{875}(93,\cdot)\)
\(\chi_{875}(107,\cdot)\)
\(\chi_{875}(207,\cdot)\)
\(\chi_{875}(268,\cdot)\)
\(\chi_{875}(282,\cdot)\)
\(\chi_{875}(368,\cdot)\)
\(\chi_{875}(382,\cdot)\)
\(\chi_{875}(457,\cdot)\)
\(\chi_{875}(543,\cdot)\)
\(\chi_{875}(618,\cdot)\)
\(\chi_{875}(632,\cdot)\)
\(\chi_{875}(718,\cdot)\)
\(\chi_{875}(732,\cdot)\)
\(\chi_{875}(793,\cdot)\)
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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 ]
\((127,626)\) → \((e\left(\frac{9}{20}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 875 }(207, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) |
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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)
