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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([0,15,4,6,16]))
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pari:[g,chi] = znchar(Mod(37937,40320))
\(\chi_{40320}(2417,\cdot)\)
\(\chi_{40320}(4433,\cdot)\)
\(\chi_{40320}(17777,\cdot)\)
\(\chi_{40320}(19793,\cdot)\)
\(\chi_{40320}(22577,\cdot)\)
\(\chi_{40320}(24593,\cdot)\)
\(\chi_{40320}(37937,\cdot)\)
\(\chi_{40320}(39953,\cdot)\)
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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 ]
\((8191,23941,17921,32257,28801)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{1}{6}\right),i,e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 40320 }(37937, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) |
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sage:chi.jacobi_sum(n)
