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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6336, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,7,0,8]))
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pari:[g,chi] = znchar(Mod(3475,6336))
\(\chi_{6336}(307,\cdot)\)
\(\chi_{6336}(1099,\cdot)\)
\(\chi_{6336}(1891,\cdot)\)
\(\chi_{6336}(2683,\cdot)\)
\(\chi_{6336}(3475,\cdot)\)
\(\chi_{6336}(4267,\cdot)\)
\(\chi_{6336}(5059,\cdot)\)
\(\chi_{6336}(5851,\cdot)\)
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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 ]
\((4159,4357,3521,1729)\) → \((-1,e\left(\frac{7}{16}\right),1,-1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 6336 }(3475, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(-i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(1\) | \(e\left(\frac{13}{16}\right)\) |
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sage:chi.jacobi_sum(n)
