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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([0,0,44]))
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pari:[g,chi] = znchar(Mod(2689,6848))
\(\chi_{6848}(193,\cdot)\)
\(\chi_{6848}(385,\cdot)\)
\(\chi_{6848}(513,\cdot)\)
\(\chi_{6848}(577,\cdot)\)
\(\chi_{6848}(897,\cdot)\)
\(\chi_{6848}(961,\cdot)\)
\(\chi_{6848}(1025,\cdot)\)
\(\chi_{6848}(1089,\cdot)\)
\(\chi_{6848}(1153,\cdot)\)
\(\chi_{6848}(1217,\cdot)\)
\(\chi_{6848}(1345,\cdot)\)
\(\chi_{6848}(1537,\cdot)\)
\(\chi_{6848}(1793,\cdot)\)
\(\chi_{6848}(1921,\cdot)\)
\(\chi_{6848}(2049,\cdot)\)
\(\chi_{6848}(2177,\cdot)\)
\(\chi_{6848}(2241,\cdot)\)
\(\chi_{6848}(2433,\cdot)\)
\(\chi_{6848}(2497,\cdot)\)
\(\chi_{6848}(2561,\cdot)\)
\(\chi_{6848}(2625,\cdot)\)
\(\chi_{6848}(2689,\cdot)\)
\(\chi_{6848}(2817,\cdot)\)
\(\chi_{6848}(2881,\cdot)\)
\(\chi_{6848}(2945,\cdot)\)
\(\chi_{6848}(3009,\cdot)\)
\(\chi_{6848}(3137,\cdot)\)
\(\chi_{6848}(3329,\cdot)\)
\(\chi_{6848}(3393,\cdot)\)
\(\chi_{6848}(3457,\cdot)\)
...
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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 ]
\((6207,1285,6529)\) → \((1,1,e\left(\frac{22}{53}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 6848 }(2689, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{53}\right)\) | \(e\left(\frac{27}{53}\right)\) | \(e\left(\frac{45}{53}\right)\) | \(e\left(\frac{6}{53}\right)\) | \(e\left(\frac{7}{53}\right)\) | \(e\left(\frac{43}{53}\right)\) | \(e\left(\frac{30}{53}\right)\) | \(e\left(\frac{2}{53}\right)\) | \(e\left(\frac{20}{53}\right)\) | \(e\left(\frac{48}{53}\right)\) |
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sage:chi.jacobi_sum(n)
