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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6223, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,24]))
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pari:[g,chi] = znchar(Mod(3558,6223))
\(\chi_{6223}(667,\cdot)\)
\(\chi_{6223}(1145,\cdot)\)
\(\chi_{6223}(1782,\cdot)\)
\(\chi_{6223}(1794,\cdot)\)
\(\chi_{6223}(2223,\cdot)\)
\(\chi_{6223}(3056,\cdot)\)
\(\chi_{6223}(3558,\cdot)\)
\(\chi_{6223}(4195,\cdot)\)
\(\chi_{6223}(4477,\cdot)\)
\(\chi_{6223}(4636,\cdot)\)
\(\chi_{6223}(5469,\cdot)\)
\(\chi_{6223}(5604,\cdot)\)
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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 ]
\((5589,638)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{4}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6223 }(3558, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) |
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sage:chi.jacobi_sum(n)
