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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([39,0,39,71]))
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pari:[g,chi] = znchar(Mod(3455,8112))
\(\chi_{8112}(95,\cdot)\)
\(\chi_{8112}(335,\cdot)\)
\(\chi_{8112}(719,\cdot)\)
\(\chi_{8112}(959,\cdot)\)
\(\chi_{8112}(1343,\cdot)\)
\(\chi_{8112}(1583,\cdot)\)
\(\chi_{8112}(1967,\cdot)\)
\(\chi_{8112}(2207,\cdot)\)
\(\chi_{8112}(2591,\cdot)\)
\(\chi_{8112}(2831,\cdot)\)
\(\chi_{8112}(3215,\cdot)\)
\(\chi_{8112}(3455,\cdot)\)
\(\chi_{8112}(3839,\cdot)\)
\(\chi_{8112}(4463,\cdot)\)
\(\chi_{8112}(4703,\cdot)\)
\(\chi_{8112}(5087,\cdot)\)
\(\chi_{8112}(5327,\cdot)\)
\(\chi_{8112}(5711,\cdot)\)
\(\chi_{8112}(5951,\cdot)\)
\(\chi_{8112}(6335,\cdot)\)
\(\chi_{8112}(6575,\cdot)\)
\(\chi_{8112}(6959,\cdot)\)
\(\chi_{8112}(7199,\cdot)\)
\(\chi_{8112}(7823,\cdot)\)
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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 ]
\((5071,6085,2705,3889)\) → \((-1,1,-1,e\left(\frac{71}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(3455, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{23}{39}\right)\) |
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sage:chi.jacobi_sum(n)
