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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([0,39,0,4]))
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pari:[g,chi] = znchar(Mod(5833,8112))
\(\chi_{8112}(217,\cdot)\)
\(\chi_{8112}(601,\cdot)\)
\(\chi_{8112}(841,\cdot)\)
\(\chi_{8112}(1225,\cdot)\)
\(\chi_{8112}(1465,\cdot)\)
\(\chi_{8112}(1849,\cdot)\)
\(\chi_{8112}(2089,\cdot)\)
\(\chi_{8112}(2473,\cdot)\)
\(\chi_{8112}(2713,\cdot)\)
\(\chi_{8112}(3097,\cdot)\)
\(\chi_{8112}(3337,\cdot)\)
\(\chi_{8112}(3721,\cdot)\)
\(\chi_{8112}(3961,\cdot)\)
\(\chi_{8112}(4345,\cdot)\)
\(\chi_{8112}(4969,\cdot)\)
\(\chi_{8112}(5209,\cdot)\)
\(\chi_{8112}(5593,\cdot)\)
\(\chi_{8112}(5833,\cdot)\)
\(\chi_{8112}(6217,\cdot)\)
\(\chi_{8112}(6457,\cdot)\)
\(\chi_{8112}(6841,\cdot)\)
\(\chi_{8112}(7081,\cdot)\)
\(\chi_{8112}(7465,\cdot)\)
\(\chi_{8112}(7705,\cdot)\)
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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 ]
\((5071,6085,2705,3889)\) → \((1,-1,1,e\left(\frac{2}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(5833, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{35}{78}\right)\) |
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sage:chi.jacobi_sum(n)
