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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2496, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,9,0,4]))
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pari:[g,chi] = znchar(Mod(1243,2496))
\(\chi_{2496}(499,\cdot)\)
\(\chi_{2496}(619,\cdot)\)
\(\chi_{2496}(1123,\cdot)\)
\(\chi_{2496}(1243,\cdot)\)
\(\chi_{2496}(1747,\cdot)\)
\(\chi_{2496}(1867,\cdot)\)
\(\chi_{2496}(2371,\cdot)\)
\(\chi_{2496}(2491,\cdot)\)
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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 ]
\((703,1093,833,769)\) → \((-1,e\left(\frac{9}{16}\right),1,i)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2496 }(1243, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(i\) | \(e\left(\frac{11}{16}\right)\) |
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sage:chi.jacobi_sum(n)
