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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12138, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([0,136,241]))
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pari:[g,chi] = znchar(Mod(139,12138))
\(\chi_{12138}(97,\cdot)\)
\(\chi_{12138}(139,\cdot)\)
\(\chi_{12138}(181,\cdot)\)
\(\chi_{12138}(265,\cdot)\)
\(\chi_{12138}(517,\cdot)\)
\(\chi_{12138}(601,\cdot)\)
\(\chi_{12138}(685,\cdot)\)
\(\chi_{12138}(811,\cdot)\)
\(\chi_{12138}(853,\cdot)\)
\(\chi_{12138}(895,\cdot)\)
\(\chi_{12138}(979,\cdot)\)
\(\chi_{12138}(1315,\cdot)\)
\(\chi_{12138}(1357,\cdot)\)
\(\chi_{12138}(1399,\cdot)\)
\(\chi_{12138}(1525,\cdot)\)
\(\chi_{12138}(1567,\cdot)\)
\(\chi_{12138}(1609,\cdot)\)
\(\chi_{12138}(1693,\cdot)\)
\(\chi_{12138}(1945,\cdot)\)
\(\chi_{12138}(2029,\cdot)\)
\(\chi_{12138}(2071,\cdot)\)
\(\chi_{12138}(2113,\cdot)\)
\(\chi_{12138}(2239,\cdot)\)
\(\chi_{12138}(2281,\cdot)\)
\(\chi_{12138}(2323,\cdot)\)
\(\chi_{12138}(2407,\cdot)\)
\(\chi_{12138}(2659,\cdot)\)
\(\chi_{12138}(2743,\cdot)\)
\(\chi_{12138}(2785,\cdot)\)
\(\chi_{12138}(2827,\cdot)\)
...
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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 ]
\((8093,10405,9829)\) → \((1,-1,e\left(\frac{241}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 12138 }(139, a) \) |
\(1\) | \(1\) | \(e\left(\frac{109}{272}\right)\) | \(e\left(\frac{103}{272}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{123}{136}\right)\) | \(e\left(\frac{159}{272}\right)\) | \(e\left(\frac{109}{136}\right)\) | \(e\left(\frac{205}{272}\right)\) | \(e\left(\frac{129}{272}\right)\) | \(e\left(\frac{81}{272}\right)\) | \(e\left(\frac{3}{272}\right)\) |
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sage:chi.jacobi_sum(n)
