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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2032, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([0,0,80]))
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pari:[g,chi] = znchar(Mod(1057,2032))
\(\chi_{2032}(17,\cdot)\)
\(\chi_{2032}(49,\cdot)\)
\(\chi_{2032}(81,\cdot)\)
\(\chi_{2032}(113,\cdot)\)
\(\chi_{2032}(145,\cdot)\)
\(\chi_{2032}(161,\cdot)\)
\(\chi_{2032}(209,\cdot)\)
\(\chi_{2032}(225,\cdot)\)
\(\chi_{2032}(289,\cdot)\)
\(\chi_{2032}(369,\cdot)\)
\(\chi_{2032}(417,\cdot)\)
\(\chi_{2032}(465,\cdot)\)
\(\chi_{2032}(529,\cdot)\)
\(\chi_{2032}(577,\cdot)\)
\(\chi_{2032}(705,\cdot)\)
\(\chi_{2032}(833,\cdot)\)
\(\chi_{2032}(961,\cdot)\)
\(\chi_{2032}(977,\cdot)\)
\(\chi_{2032}(993,\cdot)\)
\(\chi_{2032}(1009,\cdot)\)
\(\chi_{2032}(1025,\cdot)\)
\(\chi_{2032}(1057,\cdot)\)
\(\chi_{2032}(1137,\cdot)\)
\(\chi_{2032}(1169,\cdot)\)
\(\chi_{2032}(1185,\cdot)\)
\(\chi_{2032}(1217,\cdot)\)
\(\chi_{2032}(1281,\cdot)\)
\(\chi_{2032}(1441,\cdot)\)
\(\chi_{2032}(1457,\cdot)\)
\(\chi_{2032}(1521,\cdot)\)
...
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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 ]
\((255,1525,257)\) → \((1,1,e\left(\frac{40}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2032 }(1057, a) \) |
\(1\) | \(1\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{41}{63}\right)\) |
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sage:chi.jacobi_sum(n)
