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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([0,36,24,16,27]))
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pari:[g,chi] = znchar(Mod(5777,22848))
\(\chi_{22848}(3089,\cdot)\)
\(\chi_{22848}(5777,\cdot)\)
\(\chi_{22848}(6353,\cdot)\)
\(\chi_{22848}(6449,\cdot)\)
\(\chi_{22848}(7793,\cdot)\)
\(\chi_{22848}(9041,\cdot)\)
\(\chi_{22848}(9713,\cdot)\)
\(\chi_{22848}(11057,\cdot)\)
\(\chi_{22848}(11825,\cdot)\)
\(\chi_{22848}(13169,\cdot)\)
\(\chi_{22848}(13841,\cdot)\)
\(\chi_{22848}(15089,\cdot)\)
\(\chi_{22848}(16433,\cdot)\)
\(\chi_{22848}(16529,\cdot)\)
\(\chi_{22848}(17105,\cdot)\)
\(\chi_{22848}(19793,\cdot)\)
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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 ]
\((13567,18565,15233,3265,2689)\) → \((1,-i,-1,e\left(\frac{1}{3}\right),e\left(\frac{9}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 22848 }(5777, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(-1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) |
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sage:chi.jacobi_sum(n)
