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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([9,8]))
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pari:[g,chi] = znchar(Mod(18,1225))
\(\chi_{1225}(18,\cdot)\)
\(\chi_{1225}(557,\cdot)\)
\(\chi_{1225}(618,\cdot)\)
\(\chi_{1225}(1157,\cdot)\)
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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 ]
\((1177,101)\) → \((-i,e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 1225 }(18, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
