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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4005, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([0,22,57]))
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pari:[g,chi] = znchar(Mod(2782,4005))
\(\chi_{4005}(82,\cdot)\)
\(\chi_{4005}(118,\cdot)\)
\(\chi_{4005}(163,\cdot)\)
\(\chi_{4005}(172,\cdot)\)
\(\chi_{4005}(208,\cdot)\)
\(\chi_{4005}(298,\cdot)\)
\(\chi_{4005}(343,\cdot)\)
\(\chi_{4005}(397,\cdot)\)
\(\chi_{4005}(442,\cdot)\)
\(\chi_{4005}(577,\cdot)\)
\(\chi_{4005}(658,\cdot)\)
\(\chi_{4005}(847,\cdot)\)
\(\chi_{4005}(928,\cdot)\)
\(\chi_{4005}(982,\cdot)\)
\(\chi_{4005}(1027,\cdot)\)
\(\chi_{4005}(1252,\cdot)\)
\(\chi_{4005}(1342,\cdot)\)
\(\chi_{4005}(1747,\cdot)\)
\(\chi_{4005}(1972,\cdot)\)
\(\chi_{4005}(2098,\cdot)\)
\(\chi_{4005}(2197,\cdot)\)
\(\chi_{4005}(2368,\cdot)\)
\(\chi_{4005}(2557,\cdot)\)
\(\chi_{4005}(2647,\cdot)\)
\(\chi_{4005}(2683,\cdot)\)
\(\chi_{4005}(2728,\cdot)\)
\(\chi_{4005}(2782,\cdot)\)
\(\chi_{4005}(2818,\cdot)\)
\(\chi_{4005}(2863,\cdot)\)
\(\chi_{4005}(2872,\cdot)\)
...
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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 ]
\((3116,802,181)\) → \((1,i,e\left(\frac{57}{88}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4005 }(2782, a) \) |
\(1\) | \(1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{15}{88}\right)\) |
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sage:chi.jacobi_sum(n)
