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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1344, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,9,0,0]))
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pari:[g,chi] = znchar(Mod(421,1344))
\(\chi_{1344}(85,\cdot)\)
\(\chi_{1344}(253,\cdot)\)
\(\chi_{1344}(421,\cdot)\)
\(\chi_{1344}(589,\cdot)\)
\(\chi_{1344}(757,\cdot)\)
\(\chi_{1344}(925,\cdot)\)
\(\chi_{1344}(1093,\cdot)\)
\(\chi_{1344}(1261,\cdot)\)
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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 ]
\((127,1093,449,577)\) → \((1,e\left(\frac{9}{16}\right),1,1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1344 }(421, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(-1\) | \(e\left(\frac{1}{16}\right)\) |
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sage:chi.jacobi_sum(n)
