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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([57,50]))
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pari:[g,chi] = znchar(Mod(563,950))
\(\chi_{950}(27,\cdot)\)
\(\chi_{950}(103,\cdot)\)
\(\chi_{950}(183,\cdot)\)
\(\chi_{950}(217,\cdot)\)
\(\chi_{950}(297,\cdot)\)
\(\chi_{950}(373,\cdot)\)
\(\chi_{950}(483,\cdot)\)
\(\chi_{950}(487,\cdot)\)
\(\chi_{950}(563,\cdot)\)
\(\chi_{950}(597,\cdot)\)
\(\chi_{950}(673,\cdot)\)
\(\chi_{950}(677,\cdot)\)
\(\chi_{950}(753,\cdot)\)
\(\chi_{950}(787,\cdot)\)
\(\chi_{950}(863,\cdot)\)
\(\chi_{950}(867,\cdot)\)
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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 ]
\((77,401)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 950 }(563, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(-i\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) |
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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)
