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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5243, base_ring=CyclotomicField(106))
M = H._module
chi = DirichletCharacter(H, M([0,50]))
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pari:[g,chi] = znchar(Mod(1324,5243))
\(\chi_{5243}(99,\cdot)\)
\(\chi_{5243}(148,\cdot)\)
\(\chi_{5243}(197,\cdot)\)
\(\chi_{5243}(295,\cdot)\)
\(\chi_{5243}(344,\cdot)\)
\(\chi_{5243}(442,\cdot)\)
\(\chi_{5243}(785,\cdot)\)
\(\chi_{5243}(834,\cdot)\)
\(\chi_{5243}(883,\cdot)\)
\(\chi_{5243}(932,\cdot)\)
\(\chi_{5243}(1079,\cdot)\)
\(\chi_{5243}(1226,\cdot)\)
\(\chi_{5243}(1324,\cdot)\)
\(\chi_{5243}(1373,\cdot)\)
\(\chi_{5243}(1618,\cdot)\)
\(\chi_{5243}(1667,\cdot)\)
\(\chi_{5243}(1716,\cdot)\)
\(\chi_{5243}(1765,\cdot)\)
\(\chi_{5243}(1814,\cdot)\)
\(\chi_{5243}(1863,\cdot)\)
\(\chi_{5243}(1961,\cdot)\)
\(\chi_{5243}(2108,\cdot)\)
\(\chi_{5243}(2304,\cdot)\)
\(\chi_{5243}(2402,\cdot)\)
\(\chi_{5243}(2500,\cdot)\)
\(\chi_{5243}(2598,\cdot)\)
\(\chi_{5243}(2647,\cdot)\)
\(\chi_{5243}(2794,\cdot)\)
\(\chi_{5243}(2843,\cdot)\)
\(\chi_{5243}(2892,\cdot)\)
...
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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 ]
\((2355,4068)\) → \((1,e\left(\frac{25}{53}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 5243 }(1324, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{53}\right)\) | \(e\left(\frac{1}{53}\right)\) | \(e\left(\frac{50}{53}\right)\) | \(e\left(\frac{9}{53}\right)\) | \(e\left(\frac{26}{53}\right)\) | \(e\left(\frac{22}{53}\right)\) | \(e\left(\frac{2}{53}\right)\) | \(e\left(\frac{34}{53}\right)\) | \(e\left(\frac{20}{53}\right)\) | \(e\left(\frac{51}{53}\right)\) |
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sage:chi.jacobi_sum(n)
