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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8800, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,15,4]))
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pari:[g,chi] = znchar(Mod(543,8800))
\(\chi_{8800}(543,\cdot)\)
\(\chi_{8800}(2143,\cdot)\)
\(\chi_{8800}(3007,\cdot)\)
\(\chi_{8800}(3743,\cdot)\)
\(\chi_{8800}(4607,\cdot)\)
\(\chi_{8800}(6143,\cdot)\)
\(\chi_{8800}(6207,\cdot)\)
\(\chi_{8800}(8607,\cdot)\)
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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 ]
\((2751,3301,4577,5601)\) → \((-1,1,-i,e\left(\frac{1}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8800 }(543, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
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sage:chi.jacobi_sum(n)
