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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([24,45,0,40,36]))
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pari:[g,chi] = znchar(Mod(14131,22848))
\(\chi_{22848}(523,\cdot)\)
\(\chi_{22848}(2707,\cdot)\)
\(\chi_{22848}(2971,\cdot)\)
\(\chi_{22848}(5155,\cdot)\)
\(\chi_{22848}(6235,\cdot)\)
\(\chi_{22848}(8419,\cdot)\)
\(\chi_{22848}(8683,\cdot)\)
\(\chi_{22848}(10867,\cdot)\)
\(\chi_{22848}(11947,\cdot)\)
\(\chi_{22848}(14131,\cdot)\)
\(\chi_{22848}(14395,\cdot)\)
\(\chi_{22848}(16579,\cdot)\)
\(\chi_{22848}(17659,\cdot)\)
\(\chi_{22848}(19843,\cdot)\)
\(\chi_{22848}(20107,\cdot)\)
\(\chi_{22848}(22291,\cdot)\)
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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 ]
\((13567,18565,15233,3265,2689)\) → \((-1,e\left(\frac{15}{16}\right),1,e\left(\frac{5}{6}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 22848 }(14131, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{7}{8}\right)\) |
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sage:chi.jacobi_sum(n)
