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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12482, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([56]))
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pari:[g,chi] = znchar(Mod(31,12482))
\(\chi_{12482}(31,\cdot)\)
\(\chi_{12482}(439,\cdot)\)
\(\chi_{12482}(795,\cdot)\)
\(\chi_{12482}(961,\cdot)\)
\(\chi_{12482}(1905,\cdot)\)
\(\chi_{12482}(2465,\cdot)\)
\(\chi_{12482}(3209,\cdot)\)
\(\chi_{12482}(4153,\cdot)\)
\(\chi_{12482}(5491,\cdot)\)
\(\chi_{12482}(5771,\cdot)\)
\(\chi_{12482}(6845,\cdot)\)
\(\chi_{12482}(7051,\cdot)\)
\(\chi_{12482}(7925,\cdot)\)
\(\chi_{12482}(8683,\cdot)\)
\(\chi_{12482}(9079,\cdot)\)
\(\chi_{12482}(9127,\cdot)\)
\(\chi_{12482}(9245,\cdot)\)
\(\chi_{12482}(9275,\cdot)\)
\(\chi_{12482}(9595,\cdot)\)
\(\chi_{12482}(9743,\cdot)\)
\(\chi_{12482}(9767,\cdot)\)
\(\chi_{12482}(9973,\cdot)\)
\(\chi_{12482}(12163,\cdot)\)
\(\chi_{12482}(12335,\cdot)\)
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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 ]
\(3\) → \(e\left(\frac{28}{39}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 12482 }(31, a) \) |
\(1\) | \(1\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) |
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sage:chi.jacobi_sum(n)
