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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([0,39,0,53]))
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pari:[g,chi] = znchar(Mod(1609,8112))
\(\chi_{8112}(121,\cdot)\)
\(\chi_{8112}(745,\cdot)\)
\(\chi_{8112}(985,\cdot)\)
\(\chi_{8112}(1369,\cdot)\)
\(\chi_{8112}(1609,\cdot)\)
\(\chi_{8112}(1993,\cdot)\)
\(\chi_{8112}(2233,\cdot)\)
\(\chi_{8112}(2617,\cdot)\)
\(\chi_{8112}(2857,\cdot)\)
\(\chi_{8112}(3241,\cdot)\)
\(\chi_{8112}(3481,\cdot)\)
\(\chi_{8112}(4105,\cdot)\)
\(\chi_{8112}(4489,\cdot)\)
\(\chi_{8112}(4729,\cdot)\)
\(\chi_{8112}(5113,\cdot)\)
\(\chi_{8112}(5353,\cdot)\)
\(\chi_{8112}(5737,\cdot)\)
\(\chi_{8112}(5977,\cdot)\)
\(\chi_{8112}(6361,\cdot)\)
\(\chi_{8112}(6601,\cdot)\)
\(\chi_{8112}(6985,\cdot)\)
\(\chi_{8112}(7225,\cdot)\)
\(\chi_{8112}(7609,\cdot)\)
\(\chi_{8112}(7849,\cdot)\)
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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 ]
\((5071,6085,2705,3889)\) → \((1,-1,1,e\left(\frac{53}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(1609, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{25}{78}\right)\) |
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sage:chi.jacobi_sum(n)
