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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([0,1]))
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pari:[g,chi] = znchar(Mod(638,6223))
\(\chi_{6223}(393,\cdot)\)
\(\chi_{6223}(491,\cdot)\)
\(\chi_{6223}(638,\cdot)\)
\(\chi_{6223}(736,\cdot)\)
\(\chi_{6223}(785,\cdot)\)
\(\chi_{6223}(932,\cdot)\)
\(\chi_{6223}(981,\cdot)\)
\(\chi_{6223}(1030,\cdot)\)
\(\chi_{6223}(1128,\cdot)\)
\(\chi_{6223}(1226,\cdot)\)
\(\chi_{6223}(1569,\cdot)\)
\(\chi_{6223}(1716,\cdot)\)
\(\chi_{6223}(1765,\cdot)\)
\(\chi_{6223}(1863,\cdot)\)
\(\chi_{6223}(1912,\cdot)\)
\(\chi_{6223}(1961,\cdot)\)
\(\chi_{6223}(2255,\cdot)\)
\(\chi_{6223}(2353,\cdot)\)
\(\chi_{6223}(2402,\cdot)\)
\(\chi_{6223}(2598,\cdot)\)
\(\chi_{6223}(2696,\cdot)\)
\(\chi_{6223}(2745,\cdot)\)
\(\chi_{6223}(3039,\cdot)\)
\(\chi_{6223}(3284,\cdot)\)
\(\chi_{6223}(3774,\cdot)\)
\(\chi_{6223}(4117,\cdot)\)
\(\chi_{6223}(4411,\cdot)\)
\(\chi_{6223}(4705,\cdot)\)
\(\chi_{6223}(4754,\cdot)\)
\(\chi_{6223}(4999,\cdot)\)
...
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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 ]
\((5589,638)\) → \((1,e\left(\frac{1}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6223 }(638, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{19}{126}\right)\) |
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sage:chi.jacobi_sum(n)
