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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,3,8,16]))
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pari:[g,chi] = znchar(Mod(2713,4032))
\(\chi_{4032}(457,\cdot)\)
\(\chi_{4032}(697,\cdot)\)
\(\chi_{4032}(1465,\cdot)\)
\(\chi_{4032}(1705,\cdot)\)
\(\chi_{4032}(2473,\cdot)\)
\(\chi_{4032}(2713,\cdot)\)
\(\chi_{4032}(3481,\cdot)\)
\(\chi_{4032}(3721,\cdot)\)
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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 ]
\((127,3781,1793,577)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{1}{3}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 4032 }(2713, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(-i\) | \(i\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{24}\right)\) |
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sage:chi.jacobi_sum(n)
