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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6800, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,0,18,5]))
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pari:[g,chi] = znchar(Mod(2287,6800))
\(\chi_{6800}(127,\cdot)\)
\(\chi_{6800}(223,\cdot)\)
\(\chi_{6800}(927,\cdot)\)
\(\chi_{6800}(1103,\cdot)\)
\(\chi_{6800}(1487,\cdot)\)
\(\chi_{6800}(1583,\cdot)\)
\(\chi_{6800}(2287,\cdot)\)
\(\chi_{6800}(2463,\cdot)\)
\(\chi_{6800}(2847,\cdot)\)
\(\chi_{6800}(3647,\cdot)\)
\(\chi_{6800}(3823,\cdot)\)
\(\chi_{6800}(4303,\cdot)\)
\(\chi_{6800}(5183,\cdot)\)
\(\chi_{6800}(5567,\cdot)\)
\(\chi_{6800}(5663,\cdot)\)
\(\chi_{6800}(6367,\cdot)\)
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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 ]
\((5951,1701,2177,1601)\) → \((-1,1,e\left(\frac{9}{20}\right),e\left(\frac{1}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6800 }(2287, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) |
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sage:chi.jacobi_sum(n)
