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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([1,0]))
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pari:[g,chi] = znchar(Mod(5589,6223))
\(\chi_{6223}(255,\cdot)\)
\(\chi_{6223}(1144,\cdot)\)
\(\chi_{6223}(1398,\cdot)\)
\(\chi_{6223}(2033,\cdot)\)
\(\chi_{6223}(2287,\cdot)\)
\(\chi_{6223}(3176,\cdot)\)
\(\chi_{6223}(3811,\cdot)\)
\(\chi_{6223}(4065,\cdot)\)
\(\chi_{6223}(4700,\cdot)\)
\(\chi_{6223}(4954,\cdot)\)
\(\chi_{6223}(5589,\cdot)\)
\(\chi_{6223}(5843,\cdot)\)
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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 ]
\((5589,638)\) → \((e\left(\frac{1}{42}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6223 }(5589, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) |
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sage:chi.jacobi_sum(n)
