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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4998, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,0,5]))
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pari:[g,chi] = znchar(Mod(4901,4998))
\(\chi_{4998}(197,\cdot)\)
\(\chi_{4998}(785,\cdot)\)
\(\chi_{4998}(1961,\cdot)\)
\(\chi_{4998}(2255,\cdot)\)
\(\chi_{4998}(3431,\cdot)\)
\(\chi_{4998}(4019,\cdot)\)
\(\chi_{4998}(4313,\cdot)\)
\(\chi_{4998}(4901,\cdot)\)
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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 ]
\((1667,2551,4117)\) → \((-1,1,e\left(\frac{5}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4998 }(4901, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) |
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sage:chi.jacobi_sum(n)
