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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(497, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([0,48]))
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pari:[g,chi] = znchar(Mod(43,497))
\(\chi_{497}(8,\cdot)\)
\(\chi_{497}(15,\cdot)\)
\(\chi_{497}(29,\cdot)\)
\(\chi_{497}(36,\cdot)\)
\(\chi_{497}(43,\cdot)\)
\(\chi_{497}(50,\cdot)\)
\(\chi_{497}(64,\cdot)\)
\(\chi_{497}(120,\cdot)\)
\(\chi_{497}(148,\cdot)\)
\(\chi_{497}(169,\cdot)\)
\(\chi_{497}(225,\cdot)\)
\(\chi_{497}(232,\cdot)\)
\(\chi_{497}(253,\cdot)\)
\(\chi_{497}(288,\cdot)\)
\(\chi_{497}(302,\cdot)\)
\(\chi_{497}(344,\cdot)\)
\(\chi_{497}(358,\cdot)\)
\(\chi_{497}(365,\cdot)\)
\(\chi_{497}(379,\cdot)\)
\(\chi_{497}(393,\cdot)\)
\(\chi_{497}(428,\cdot)\)
\(\chi_{497}(435,\cdot)\)
\(\chi_{497}(442,\cdot)\)
\(\chi_{497}(484,\cdot)\)
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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 ]
\((143,78)\) → \((1,e\left(\frac{24}{35}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 497 }(43, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) |
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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)
