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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2349, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([35,12]))
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pari:[g,chi] = znchar(Mod(53,2349))
\(\chi_{2349}(53,\cdot)\)
\(\chi_{2349}(107,\cdot)\)
\(\chi_{2349}(431,\cdot)\)
\(\chi_{2349}(458,\cdot)\)
\(\chi_{2349}(674,\cdot)\)
\(\chi_{2349}(836,\cdot)\)
\(\chi_{2349}(944,\cdot)\)
\(\chi_{2349}(1241,\cdot)\)
\(\chi_{2349}(1673,\cdot)\)
\(\chi_{2349}(1727,\cdot)\)
\(\chi_{2349}(1997,\cdot)\)
\(\chi_{2349}(2240,\cdot)\)
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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 ]
\((407,1945)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{2}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(53, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) |
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sage:chi.jacobi_sum(n)
