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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2349, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([28,57]))
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pari:[g,chi] = znchar(Mod(838,2349))
\(\chi_{2349}(55,\cdot)\)
\(\chi_{2349}(217,\cdot)\)
\(\chi_{2349}(271,\cdot)\)
\(\chi_{2349}(298,\cdot)\)
\(\chi_{2349}(379,\cdot)\)
\(\chi_{2349}(433,\cdot)\)
\(\chi_{2349}(514,\cdot)\)
\(\chi_{2349}(541,\cdot)\)
\(\chi_{2349}(595,\cdot)\)
\(\chi_{2349}(757,\cdot)\)
\(\chi_{2349}(838,\cdot)\)
\(\chi_{2349}(946,\cdot)\)
\(\chi_{2349}(1000,\cdot)\)
\(\chi_{2349}(1081,\cdot)\)
\(\chi_{2349}(1162,\cdot)\)
\(\chi_{2349}(1324,\cdot)\)
\(\chi_{2349}(1432,\cdot)\)
\(\chi_{2349}(1729,\cdot)\)
\(\chi_{2349}(1837,\cdot)\)
\(\chi_{2349}(1999,\cdot)\)
\(\chi_{2349}(2080,\cdot)\)
\(\chi_{2349}(2161,\cdot)\)
\(\chi_{2349}(2215,\cdot)\)
\(\chi_{2349}(2323,\cdot)\)
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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 ]
\((407,1945)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{19}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(838, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{1}{21}\right)\) |
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sage:chi.jacobi_sum(n)
