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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4080, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,4,0,4,13]))
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pari:[g,chi] = znchar(Mod(3157,4080))
\(\chi_{4080}(37,\cdot)\)
\(\chi_{4080}(277,\cdot)\)
\(\chi_{4080}(1213,\cdot)\)
\(\chi_{4080}(1693,\cdot)\)
\(\chi_{4080}(3157,\cdot)\)
\(\chi_{4080}(3373,\cdot)\)
\(\chi_{4080}(3397,\cdot)\)
\(\chi_{4080}(3853,\cdot)\)
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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 ]
\((511,3061,1361,817,241)\) → \((1,i,1,i,e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4080 }(3157, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) |
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sage:chi.jacobi_sum(n)
