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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(381, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([0,82]))
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pari:[g,chi] = znchar(Mod(115,381))
\(\chi_{381}(13,\cdot)\)
\(\chi_{381}(31,\cdot)\)
\(\chi_{381}(34,\cdot)\)
\(\chi_{381}(49,\cdot)\)
\(\chi_{381}(70,\cdot)\)
\(\chi_{381}(79,\cdot)\)
\(\chi_{381}(82,\cdot)\)
\(\chi_{381}(88,\cdot)\)
\(\chi_{381}(115,\cdot)\)
\(\chi_{381}(121,\cdot)\)
\(\chi_{381}(124,\cdot)\)
\(\chi_{381}(136,\cdot)\)
\(\chi_{381}(142,\cdot)\)
\(\chi_{381}(145,\cdot)\)
\(\chi_{381}(148,\cdot)\)
\(\chi_{381}(157,\cdot)\)
\(\chi_{381}(163,\cdot)\)
\(\chi_{381}(169,\cdot)\)
\(\chi_{381}(187,\cdot)\)
\(\chi_{381}(196,\cdot)\)
\(\chi_{381}(199,\cdot)\)
\(\chi_{381}(208,\cdot)\)
\(\chi_{381}(211,\cdot)\)
\(\chi_{381}(247,\cdot)\)
\(\chi_{381}(265,\cdot)\)
\(\chi_{381}(271,\cdot)\)
\(\chi_{381}(280,\cdot)\)
\(\chi_{381}(289,\cdot)\)
\(\chi_{381}(295,\cdot)\)
\(\chi_{381}(298,\cdot)\)
...
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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 ]
\((128,130)\) → \((1,e\left(\frac{41}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 381 }(115, a) \) |
\(1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) |
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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)
