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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2380, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,18,4,3]))
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pari:[g,chi] = znchar(Mod(213,2380))
\(\chi_{2380}(117,\cdot)\)
\(\chi_{2380}(213,\cdot)\)
\(\chi_{2380}(257,\cdot)\)
\(\chi_{2380}(773,\cdot)\)
\(\chi_{2380}(1137,\cdot)\)
\(\chi_{2380}(1277,\cdot)\)
\(\chi_{2380}(1573,\cdot)\)
\(\chi_{2380}(2133,\cdot)\)
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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 ]
\((1191,477,1361,1261)\) → \((1,-i,e\left(\frac{1}{6}\right),e\left(\frac{1}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(33\) |
| \( \chi_{ 2380 }(213, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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sage:chi.jacobi_sum(n)
