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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6848, base_ring=CyclotomicField(106))
M = H._module
chi = DirichletCharacter(H, M([53,0,21]))
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pari:[g,chi] = znchar(Mod(1343,6848))
\(\chi_{6848}(63,\cdot)\)
\(\chi_{6848}(127,\cdot)\)
\(\chi_{6848}(191,\cdot)\)
\(\chi_{6848}(639,\cdot)\)
\(\chi_{6848}(767,\cdot)\)
\(\chi_{6848}(831,\cdot)\)
\(\chi_{6848}(959,\cdot)\)
\(\chi_{6848}(1023,\cdot)\)
\(\chi_{6848}(1087,\cdot)\)
\(\chi_{6848}(1215,\cdot)\)
\(\chi_{6848}(1343,\cdot)\)
\(\chi_{6848}(1471,\cdot)\)
\(\chi_{6848}(1663,\cdot)\)
\(\chi_{6848}(1727,\cdot)\)
\(\chi_{6848}(2111,\cdot)\)
\(\chi_{6848}(2431,\cdot)\)
\(\chi_{6848}(2559,\cdot)\)
\(\chi_{6848}(2623,\cdot)\)
\(\chi_{6848}(2879,\cdot)\)
\(\chi_{6848}(2943,\cdot)\)
\(\chi_{6848}(3135,\cdot)\)
\(\chi_{6848}(3199,\cdot)\)
\(\chi_{6848}(3391,\cdot)\)
\(\chi_{6848}(3455,\cdot)\)
\(\chi_{6848}(3519,\cdot)\)
\(\chi_{6848}(3711,\cdot)\)
\(\chi_{6848}(3839,\cdot)\)
\(\chi_{6848}(3903,\cdot)\)
\(\chi_{6848}(3967,\cdot)\)
\(\chi_{6848}(4031,\cdot)\)
...
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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 ]
\((6207,1285,6529)\) → \((-1,1,e\left(\frac{21}{106}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 6848 }(1343, a) \) |
\(1\) | \(1\) | \(e\left(\frac{39}{106}\right)\) | \(e\left(\frac{33}{106}\right)\) | \(e\left(\frac{1}{53}\right)\) | \(e\left(\frac{39}{53}\right)\) | \(e\left(\frac{91}{106}\right)\) | \(e\left(\frac{41}{53}\right)\) | \(e\left(\frac{36}{53}\right)\) | \(e\left(\frac{79}{106}\right)\) | \(e\left(\frac{101}{106}\right)\) | \(e\left(\frac{41}{106}\right)\) |
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sage:chi.jacobi_sum(n)
