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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40320, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,15,20,18,4]))
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pari:[g,chi] = znchar(Mod(6863,40320))
\(\chi_{40320}(47,\cdot)\)
\(\chi_{40320}(2063,\cdot)\)
\(\chi_{40320}(4847,\cdot)\)
\(\chi_{40320}(6863,\cdot)\)
\(\chi_{40320}(20207,\cdot)\)
\(\chi_{40320}(22223,\cdot)\)
\(\chi_{40320}(25007,\cdot)\)
\(\chi_{40320}(27023,\cdot)\)
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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 ]
\((8191,23941,17921,32257,28801)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{5}{6}\right),-i,e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 40320 }(6863, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) |
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sage:chi.jacobi_sum(n)
