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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(296, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,0,1]))
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pari:[g,chi] = znchar(Mod(39,296))
\(\chi_{296}(15,\cdot)\)
\(\chi_{296}(39,\cdot)\)
\(\chi_{296}(55,\cdot)\)
\(\chi_{296}(79,\cdot)\)
\(\chi_{296}(87,\cdot)\)
\(\chi_{296}(135,\cdot)\)
\(\chi_{296}(143,\cdot)\)
\(\chi_{296}(167,\cdot)\)
\(\chi_{296}(183,\cdot)\)
\(\chi_{296}(207,\cdot)\)
\(\chi_{296}(239,\cdot)\)
\(\chi_{296}(279,\cdot)\)
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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 ]
\((223,149,113)\) → \((-1,1,e\left(\frac{1}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 296 }(39, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) |
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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)
