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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1122, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,8,13]))
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pari:[g,chi] = znchar(Mod(505,1122))
\(\chi_{1122}(109,\cdot)\)
\(\chi_{1122}(175,\cdot)\)
\(\chi_{1122}(241,\cdot)\)
\(\chi_{1122}(439,\cdot)\)
\(\chi_{1122}(505,\cdot)\)
\(\chi_{1122}(571,\cdot)\)
\(\chi_{1122}(703,\cdot)\)
\(\chi_{1122}(1099,\cdot)\)
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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 ]
\((749,409,1057)\) → \((1,-1,e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 1122 }(505, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(-1\) | \(e\left(\frac{13}{16}\right)\) |
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sage:chi.jacobi_sum(n)
