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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2496, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,3,12,8]))
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pari:[g,chi] = znchar(Mod(1511,2496))
\(\chi_{2496}(263,\cdot)\)
\(\chi_{2496}(503,\cdot)\)
\(\chi_{2496}(887,\cdot)\)
\(\chi_{2496}(1127,\cdot)\)
\(\chi_{2496}(1511,\cdot)\)
\(\chi_{2496}(1751,\cdot)\)
\(\chi_{2496}(2135,\cdot)\)
\(\chi_{2496}(2375,\cdot)\)
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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 ]
\((703,1093,833,769)\) → \((-1,e\left(\frac{1}{8}\right),-1,e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2496 }(1511, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(i\) | \(e\left(\frac{5}{24}\right)\) | \(-1\) | \(e\left(\frac{1}{24}\right)\) |
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sage:chi.jacobi_sum(n)
