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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([24,12,24,40,3]))
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pari:[g,chi] = znchar(Mod(20879,22848))
\(\chi_{22848}(719,\cdot)\)
\(\chi_{22848}(6863,\cdot)\)
\(\chi_{22848}(9551,\cdot)\)
\(\chi_{22848}(10127,\cdot)\)
\(\chi_{22848}(10223,\cdot)\)
\(\chi_{22848}(11567,\cdot)\)
\(\chi_{22848}(12815,\cdot)\)
\(\chi_{22848}(13487,\cdot)\)
\(\chi_{22848}(14831,\cdot)\)
\(\chi_{22848}(15599,\cdot)\)
\(\chi_{22848}(16943,\cdot)\)
\(\chi_{22848}(17615,\cdot)\)
\(\chi_{22848}(18863,\cdot)\)
\(\chi_{22848}(20207,\cdot)\)
\(\chi_{22848}(20303,\cdot)\)
\(\chi_{22848}(20879,\cdot)\)
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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 ]
\((13567,18565,15233,3265,2689)\) → \((-1,i,-1,e\left(\frac{5}{6}\right),e\left(\frac{1}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 22848 }(20879, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(-1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) |
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sage:chi.jacobi_sum(n)
