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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3888, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([0,81,80]))
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pari:[g,chi] = znchar(Mod(2557,3888))
\(\chi_{3888}(37,\cdot)\)
\(\chi_{3888}(181,\cdot)\)
\(\chi_{3888}(253,\cdot)\)
\(\chi_{3888}(397,\cdot)\)
\(\chi_{3888}(469,\cdot)\)
\(\chi_{3888}(613,\cdot)\)
\(\chi_{3888}(685,\cdot)\)
\(\chi_{3888}(829,\cdot)\)
\(\chi_{3888}(901,\cdot)\)
\(\chi_{3888}(1045,\cdot)\)
\(\chi_{3888}(1117,\cdot)\)
\(\chi_{3888}(1261,\cdot)\)
\(\chi_{3888}(1333,\cdot)\)
\(\chi_{3888}(1477,\cdot)\)
\(\chi_{3888}(1549,\cdot)\)
\(\chi_{3888}(1693,\cdot)\)
\(\chi_{3888}(1765,\cdot)\)
\(\chi_{3888}(1909,\cdot)\)
\(\chi_{3888}(1981,\cdot)\)
\(\chi_{3888}(2125,\cdot)\)
\(\chi_{3888}(2197,\cdot)\)
\(\chi_{3888}(2341,\cdot)\)
\(\chi_{3888}(2413,\cdot)\)
\(\chi_{3888}(2557,\cdot)\)
\(\chi_{3888}(2629,\cdot)\)
\(\chi_{3888}(2773,\cdot)\)
\(\chi_{3888}(2845,\cdot)\)
\(\chi_{3888}(2989,\cdot)\)
\(\chi_{3888}(3061,\cdot)\)
\(\chi_{3888}(3205,\cdot)\)
...
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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 ]
\((2431,2917,1217)\) → \((1,-i,e\left(\frac{20}{27}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 3888 }(2557, a) \) |
\(1\) | \(1\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{22}{27}\right)\) |
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sage:chi.jacobi_sum(n)
