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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6800, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,11,15]))
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pari:[g,chi] = znchar(Mod(1823,6800))
\(\chi_{6800}(47,\cdot)\)
\(\chi_{6800}(463,\cdot)\)
\(\chi_{6800}(1823,\cdot)\)
\(\chi_{6800}(2767,\cdot)\)
\(\chi_{6800}(3183,\cdot)\)
\(\chi_{6800}(4127,\cdot)\)
\(\chi_{6800}(5487,\cdot)\)
\(\chi_{6800}(5903,\cdot)\)
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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 ]
\((5951,1701,2177,1601)\) → \((-1,1,e\left(\frac{11}{20}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6800 }(1823, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(-1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) |
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sage:chi.jacobi_sum(n)
