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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(429, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,42,35]))
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pari:[g,chi] = znchar(Mod(271,429))
\(\chi_{429}(7,\cdot)\)
\(\chi_{429}(19,\cdot)\)
\(\chi_{429}(28,\cdot)\)
\(\chi_{429}(46,\cdot)\)
\(\chi_{429}(85,\cdot)\)
\(\chi_{429}(106,\cdot)\)
\(\chi_{429}(145,\cdot)\)
\(\chi_{429}(184,\cdot)\)
\(\chi_{429}(193,\cdot)\)
\(\chi_{429}(271,\cdot)\)
\(\chi_{429}(292,\cdot)\)
\(\chi_{429}(310,\cdot)\)
\(\chi_{429}(349,\cdot)\)
\(\chi_{429}(358,\cdot)\)
\(\chi_{429}(370,\cdot)\)
\(\chi_{429}(409,\cdot)\)
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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 ]
\((287,79,67)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{7}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 429 }(271, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) |
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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)
