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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(980, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,31]))
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pari:[g,chi] = znchar(Mod(829,980))
\(\chi_{980}(89,\cdot)\)
\(\chi_{980}(229,\cdot)\)
\(\chi_{980}(269,\cdot)\)
\(\chi_{980}(369,\cdot)\)
\(\chi_{980}(409,\cdot)\)
\(\chi_{980}(549,\cdot)\)
\(\chi_{980}(649,\cdot)\)
\(\chi_{980}(689,\cdot)\)
\(\chi_{980}(789,\cdot)\)
\(\chi_{980}(829,\cdot)\)
\(\chi_{980}(929,\cdot)\)
\(\chi_{980}(969,\cdot)\)
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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 ]
\((491,197,101)\) → \((1,-1,e\left(\frac{31}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 980 }(829, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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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)
