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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([0,24,24,32,9]))
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pari:[g,chi] = znchar(Mod(9377,22848))
\(\chi_{22848}(737,\cdot)\)
\(\chi_{22848}(2081,\cdot)\)
\(\chi_{22848}(2657,\cdot)\)
\(\chi_{22848}(4001,\cdot)\)
\(\chi_{22848}(5345,\cdot)\)
\(\chi_{22848}(6113,\cdot)\)
\(\chi_{22848}(7457,\cdot)\)
\(\chi_{22848}(8801,\cdot)\)
\(\chi_{22848}(9377,\cdot)\)
\(\chi_{22848}(10721,\cdot)\)
\(\chi_{22848}(11489,\cdot)\)
\(\chi_{22848}(12065,\cdot)\)
\(\chi_{22848}(14753,\cdot)\)
\(\chi_{22848}(19553,\cdot)\)
\(\chi_{22848}(22241,\cdot)\)
\(\chi_{22848}(22817,\cdot)\)
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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 ]
\((13567,18565,15233,3265,2689)\) → \((1,-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 22848 }(9377, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(i\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{9}{16}\right)\) |
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sage:chi.jacobi_sum(n)
