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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6223, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([84,80]))
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pari:[g,chi] = znchar(Mod(4232,6223))
\(\chi_{6223}(18,\cdot)\)
\(\chi_{6223}(30,\cdot)\)
\(\chi_{6223}(79,\cdot)\)
\(\chi_{6223}(263,\cdot)\)
\(\chi_{6223}(324,\cdot)\)
\(\chi_{6223}(557,\cdot)\)
\(\chi_{6223}(606,\cdot)\)
\(\chi_{6223}(704,\cdot)\)
\(\chi_{6223}(716,\cdot)\)
\(\chi_{6223}(900,\cdot)\)
\(\chi_{6223}(1010,\cdot)\)
\(\chi_{6223}(1439,\cdot)\)
\(\chi_{6223}(1733,\cdot)\)
\(\chi_{6223}(2076,\cdot)\)
\(\chi_{6223}(2370,\cdot)\)
\(\chi_{6223}(2811,\cdot)\)
\(\chi_{6223}(3362,\cdot)\)
\(\chi_{6223}(3460,\cdot)\)
\(\chi_{6223}(3644,\cdot)\)
\(\chi_{6223}(3754,\cdot)\)
\(\chi_{6223}(3803,\cdot)\)
\(\chi_{6223}(3950,\cdot)\)
\(\chi_{6223}(3999,\cdot)\)
\(\chi_{6223}(4085,\cdot)\)
\(\chi_{6223}(4232,\cdot)\)
\(\chi_{6223}(4587,\cdot)\)
\(\chi_{6223}(4685,\cdot)\)
\(\chi_{6223}(4734,\cdot)\)
\(\chi_{6223}(4771,\cdot)\)
\(\chi_{6223}(4930,\cdot)\)
...
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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 ]
\((5589,638)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{40}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6223 }(4232, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) |
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sage:chi.jacobi_sum(n)
