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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6848, base_ring=CyclotomicField(212))
M = H._module
chi = DirichletCharacter(H, M([0,53,172]))
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pari:[g,chi] = znchar(Mod(49,6848))
\(\chi_{6848}(49,\cdot)\)
\(\chi_{6848}(81,\cdot)\)
\(\chi_{6848}(209,\cdot)\)
\(\chi_{6848}(241,\cdot)\)
\(\chi_{6848}(337,\cdot)\)
\(\chi_{6848}(369,\cdot)\)
\(\chi_{6848}(465,\cdot)\)
\(\chi_{6848}(497,\cdot)\)
\(\chi_{6848}(529,\cdot)\)
\(\chi_{6848}(625,\cdot)\)
\(\chi_{6848}(689,\cdot)\)
\(\chi_{6848}(721,\cdot)\)
\(\chi_{6848}(753,\cdot)\)
\(\chi_{6848}(785,\cdot)\)
\(\chi_{6848}(849,\cdot)\)
\(\chi_{6848}(881,\cdot)\)
\(\chi_{6848}(913,\cdot)\)
\(\chi_{6848}(945,\cdot)\)
\(\chi_{6848}(977,\cdot)\)
\(\chi_{6848}(1073,\cdot)\)
\(\chi_{6848}(1105,\cdot)\)
\(\chi_{6848}(1169,\cdot)\)
\(\chi_{6848}(1233,\cdot)\)
\(\chi_{6848}(1297,\cdot)\)
\(\chi_{6848}(1425,\cdot)\)
\(\chi_{6848}(1521,\cdot)\)
\(\chi_{6848}(1585,\cdot)\)
\(\chi_{6848}(1617,\cdot)\)
\(\chi_{6848}(1649,\cdot)\)
\(\chi_{6848}(1681,\cdot)\)
...
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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 ]
\((6207,1285,6529)\) → \((1,i,e\left(\frac{43}{53}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 6848 }(49, a) \) |
\(1\) | \(1\) | \(e\left(\frac{115}{212}\right)\) | \(e\left(\frac{81}{212}\right)\) | \(e\left(\frac{41}{106}\right)\) | \(e\left(\frac{9}{106}\right)\) | \(e\left(\frac{21}{212}\right)\) | \(e\left(\frac{23}{212}\right)\) | \(e\left(\frac{49}{53}\right)\) | \(e\left(\frac{28}{53}\right)\) | \(e\left(\frac{7}{212}\right)\) | \(e\left(\frac{197}{212}\right)\) |
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sage:chi.jacobi_sum(n)
