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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1600, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,13,4]))
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pari:[g,chi] = znchar(Mod(107,1600))
\(\chi_{1600}(107,\cdot)\)
\(\chi_{1600}(243,\cdot)\)
\(\chi_{1600}(507,\cdot)\)
\(\chi_{1600}(643,\cdot)\)
\(\chi_{1600}(907,\cdot)\)
\(\chi_{1600}(1043,\cdot)\)
\(\chi_{1600}(1307,\cdot)\)
\(\chi_{1600}(1443,\cdot)\)
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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 ]
\((1151,901,577)\) → \((-1,e\left(\frac{13}{16}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1600 }(107, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) |
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sage:chi.jacobi_sum(n)
