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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(490, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,4]))
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pari:[g,chi] = znchar(Mod(81,490))
\(\chi_{490}(11,\cdot)\)
\(\chi_{490}(51,\cdot)\)
\(\chi_{490}(81,\cdot)\)
\(\chi_{490}(121,\cdot)\)
\(\chi_{490}(151,\cdot)\)
\(\chi_{490}(191,\cdot)\)
\(\chi_{490}(221,\cdot)\)
\(\chi_{490}(261,\cdot)\)
\(\chi_{490}(291,\cdot)\)
\(\chi_{490}(331,\cdot)\)
\(\chi_{490}(401,\cdot)\)
\(\chi_{490}(431,\cdot)\)
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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 ]
\((197,101)\) → \((1,e\left(\frac{2}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 490 }(81, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
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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)
