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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4998, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,25,0]))
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pari:[g,chi] = znchar(Mod(2075,4998))
\(\chi_{4998}(341,\cdot)\)
\(\chi_{4998}(647,\cdot)\)
\(\chi_{4998}(1055,\cdot)\)
\(\chi_{4998}(1361,\cdot)\)
\(\chi_{4998}(1769,\cdot)\)
\(\chi_{4998}(2075,\cdot)\)
\(\chi_{4998}(2483,\cdot)\)
\(\chi_{4998}(2789,\cdot)\)
\(\chi_{4998}(3197,\cdot)\)
\(\chi_{4998}(3503,\cdot)\)
\(\chi_{4998}(3911,\cdot)\)
\(\chi_{4998}(4217,\cdot)\)
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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 ]
\((1667,2551,4117)\) → \((-1,e\left(\frac{25}{42}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4998 }(2075, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) |
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sage:chi.jacobi_sum(n)
