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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7803, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([8,9]))
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pari:[g,chi] = znchar(Mod(712,7803))
\(\chi_{7803}(712,\cdot)\)
\(\chi_{7803}(2422,\cdot)\)
\(\chi_{7803}(2467,\cdot)\)
\(\chi_{7803}(3313,\cdot)\)
\(\chi_{7803}(3358,\cdot)\)
\(\chi_{7803}(5068,\cdot)\)
\(\chi_{7803}(5959,\cdot)\)
\(\chi_{7803}(7624,\cdot)\)
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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 ]
\((2891,2026)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 7803 }(712, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
