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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(676, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,20]))
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pari:[g,chi] = znchar(Mod(53,676))
\(\chi_{676}(53,\cdot)\)
\(\chi_{676}(105,\cdot)\)
\(\chi_{676}(157,\cdot)\)
\(\chi_{676}(209,\cdot)\)
\(\chi_{676}(261,\cdot)\)
\(\chi_{676}(313,\cdot)\)
\(\chi_{676}(365,\cdot)\)
\(\chi_{676}(417,\cdot)\)
\(\chi_{676}(469,\cdot)\)
\(\chi_{676}(521,\cdot)\)
\(\chi_{676}(573,\cdot)\)
\(\chi_{676}(625,\cdot)\)
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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 ]
\((339,509)\) → \((1,e\left(\frac{10}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 676 }(53, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(1\) |
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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)
