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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,13,0,6]))
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pari:[g,chi] = znchar(Mod(4681,8112))
\(\chi_{8112}(313,\cdot)\)
\(\chi_{8112}(937,\cdot)\)
\(\chi_{8112}(1561,\cdot)\)
\(\chi_{8112}(2185,\cdot)\)
\(\chi_{8112}(2809,\cdot)\)
\(\chi_{8112}(3433,\cdot)\)
\(\chi_{8112}(4681,\cdot)\)
\(\chi_{8112}(5305,\cdot)\)
\(\chi_{8112}(5929,\cdot)\)
\(\chi_{8112}(6553,\cdot)\)
\(\chi_{8112}(7177,\cdot)\)
\(\chi_{8112}(7801,\cdot)\)
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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 ]
\((5071,6085,2705,3889)\) → \((1,-1,1,e\left(\frac{3}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(4681, a) \) |
\(1\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) |
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sage:chi.jacobi_sum(n)
