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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(459, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([40,9]))
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pari:[g,chi] = znchar(Mod(44,459))
\(\chi_{459}(44,\cdot)\)
\(\chi_{459}(62,\cdot)\)
\(\chi_{459}(71,\cdot)\)
\(\chi_{459}(116,\cdot)\)
\(\chi_{459}(125,\cdot)\)
\(\chi_{459}(143,\cdot)\)
\(\chi_{459}(197,\cdot)\)
\(\chi_{459}(224,\cdot)\)
\(\chi_{459}(233,\cdot)\)
\(\chi_{459}(260,\cdot)\)
\(\chi_{459}(278,\cdot)\)
\(\chi_{459}(368,\cdot)\)
\(\chi_{459}(386,\cdot)\)
\(\chi_{459}(413,\cdot)\)
\(\chi_{459}(422,\cdot)\)
\(\chi_{459}(449,\cdot)\)
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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 ]
\((137,190)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 459 }(44, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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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)
