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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7742, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([26,44]))
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pari:[g,chi] = znchar(Mod(863,7742))
\(\chi_{7742}(177,\cdot)\)
\(\chi_{7742}(361,\cdot)\)
\(\chi_{7742}(753,\cdot)\)
\(\chi_{7742}(863,\cdot)\)
\(\chi_{7742}(1047,\cdot)\)
\(\chi_{7742}(1059,\cdot)\)
\(\chi_{7742}(1537,\cdot)\)
\(\chi_{7742}(1843,\cdot)\)
\(\chi_{7742}(2137,\cdot)\)
\(\chi_{7742}(2419,\cdot)\)
\(\chi_{7742}(3399,\cdot)\)
\(\chi_{7742}(3803,\cdot)\)
\(\chi_{7742}(4587,\cdot)\)
\(\chi_{7742}(5175,\cdot)\)
\(\chi_{7742}(5555,\cdot)\)
\(\chi_{7742}(5653,\cdot)\)
\(\chi_{7742}(6155,\cdot)\)
\(\chi_{7742}(6351,\cdot)\)
\(\chi_{7742}(6449,\cdot)\)
\(\chi_{7742}(6633,\cdot)\)
\(\chi_{7742}(6645,\cdot)\)
\(\chi_{7742}(6731,\cdot)\)
\(\chi_{7742}(7123,\cdot)\)
\(\chi_{7742}(7319,\cdot)\)
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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 ]
\((2845,4901)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{22}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 7742 }(863, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{39}\right)\) |
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sage:chi.jacobi_sum(n)
