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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,37]))
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pari:[g,chi] = znchar(Mod(15,287))
\(\chi_{287}(15,\cdot)\)
\(\chi_{287}(22,\cdot)\)
\(\chi_{287}(29,\cdot)\)
\(\chi_{287}(71,\cdot)\)
\(\chi_{287}(99,\cdot)\)
\(\chi_{287}(106,\cdot)\)
\(\chi_{287}(134,\cdot)\)
\(\chi_{287}(176,\cdot)\)
\(\chi_{287}(183,\cdot)\)
\(\chi_{287}(190,\cdot)\)
\(\chi_{287}(211,\cdot)\)
\(\chi_{287}(218,\cdot)\)
\(\chi_{287}(239,\cdot)\)
\(\chi_{287}(253,\cdot)\)
\(\chi_{287}(274,\cdot)\)
\(\chi_{287}(281,\cdot)\)
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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 ]
\((206,211)\) → \((1,e\left(\frac{37}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 287 }(15, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(-i\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) |
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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)
