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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2349, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([7,0]))
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pari:[g,chi] = znchar(Mod(1277,2349))
\(\chi_{2349}(233,\cdot)\)
\(\chi_{2349}(494,\cdot)\)
\(\chi_{2349}(1016,\cdot)\)
\(\chi_{2349}(1277,\cdot)\)
\(\chi_{2349}(1799,\cdot)\)
\(\chi_{2349}(2060,\cdot)\)
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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 ]
\((407,1945)\) → \((e\left(\frac{7}{18}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(1277, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) |
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sage:chi.jacobi_sum(n)
