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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([0,20,24,35]))
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pari:[g,chi] = znchar(Mod(121,6800))
\(\chi_{6800}(121,\cdot)\)
\(\chi_{6800}(281,\cdot)\)
\(\chi_{6800}(841,\cdot)\)
\(\chi_{6800}(1481,\cdot)\)
\(\chi_{6800}(1641,\cdot)\)
\(\chi_{6800}(2361,\cdot)\)
\(\chi_{6800}(2841,\cdot)\)
\(\chi_{6800}(3561,\cdot)\)
\(\chi_{6800}(3721,\cdot)\)
\(\chi_{6800}(4361,\cdot)\)
\(\chi_{6800}(4921,\cdot)\)
\(\chi_{6800}(5081,\cdot)\)
\(\chi_{6800}(5561,\cdot)\)
\(\chi_{6800}(5721,\cdot)\)
\(\chi_{6800}(6281,\cdot)\)
\(\chi_{6800}(6441,\cdot)\)
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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 ]
\((5951,1701,2177,1601)\) → \((1,-1,e\left(\frac{3}{5}\right),e\left(\frac{7}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6800 }(121, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) |
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sage:chi.jacobi_sum(n)
