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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([21,46]))
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pari:[g,chi] = znchar(Mod(31,6223))
\(\chi_{6223}(31,\cdot)\)
\(\chi_{6223}(215,\cdot)\)
\(\chi_{6223}(325,\cdot)\)
\(\chi_{6223}(374,\cdot)\)
\(\chi_{6223}(521,\cdot)\)
\(\chi_{6223}(570,\cdot)\)
\(\chi_{6223}(656,\cdot)\)
\(\chi_{6223}(803,\cdot)\)
\(\chi_{6223}(1158,\cdot)\)
\(\chi_{6223}(1256,\cdot)\)
\(\chi_{6223}(1305,\cdot)\)
\(\chi_{6223}(1342,\cdot)\)
\(\chi_{6223}(1501,\cdot)\)
\(\chi_{6223}(1550,\cdot)\)
\(\chi_{6223}(1648,\cdot)\)
\(\chi_{6223}(1685,\cdot)\)
\(\chi_{6223}(1893,\cdot)\)
\(\chi_{6223}(1979,\cdot)\)
\(\chi_{6223}(2322,\cdot)\)
\(\chi_{6223}(2812,\cdot)\)
\(\chi_{6223}(2824,\cdot)\)
\(\chi_{6223}(2873,\cdot)\)
\(\chi_{6223}(3057,\cdot)\)
\(\chi_{6223}(3118,\cdot)\)
\(\chi_{6223}(3351,\cdot)\)
\(\chi_{6223}(3400,\cdot)\)
\(\chi_{6223}(3498,\cdot)\)
\(\chi_{6223}(3510,\cdot)\)
\(\chi_{6223}(3694,\cdot)\)
\(\chi_{6223}(3804,\cdot)\)
...
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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 ]
\((5589,638)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{23}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6223 }(31, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{97}{126}\right)\) |
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sage:chi.jacobi_sum(n)
