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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22848, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,15,0,8,42]))
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pari:[g,chi] = znchar(Mod(5323,22848))
\(\chi_{22848}(451,\cdot)\)
\(\chi_{22848}(3715,\cdot)\)
\(\chi_{22848}(4819,\cdot)\)
\(\chi_{22848}(5323,\cdot)\)
\(\chi_{22848}(5659,\cdot)\)
\(\chi_{22848}(8083,\cdot)\)
\(\chi_{22848}(8587,\cdot)\)
\(\chi_{22848}(8923,\cdot)\)
\(\chi_{22848}(11875,\cdot)\)
\(\chi_{22848}(15139,\cdot)\)
\(\chi_{22848}(16243,\cdot)\)
\(\chi_{22848}(16747,\cdot)\)
\(\chi_{22848}(17083,\cdot)\)
\(\chi_{22848}(19507,\cdot)\)
\(\chi_{22848}(20011,\cdot)\)
\(\chi_{22848}(20347,\cdot)\)
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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 ]
\((13567,18565,15233,3265,2689)\) → \((-1,e\left(\frac{5}{16}\right),1,e\left(\frac{1}{6}\right),e\left(\frac{7}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 22848 }(5323, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
