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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5610, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,60,64,5]))
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pari:[g,chi] = znchar(Mod(1873,5610))
\(\chi_{5610}(367,\cdot)\)
\(\chi_{5610}(487,\cdot)\)
\(\chi_{5610}(643,\cdot)\)
\(\chi_{5610}(907,\cdot)\)
\(\chi_{5610}(1027,\cdot)\)
\(\chi_{5610}(1093,\cdot)\)
\(\chi_{5610}(1153,\cdot)\)
\(\chi_{5610}(1417,\cdot)\)
\(\chi_{5610}(1423,\cdot)\)
\(\chi_{5610}(1873,\cdot)\)
\(\chi_{5610}(1897,\cdot)\)
\(\chi_{5610}(2017,\cdot)\)
\(\chi_{5610}(2407,\cdot)\)
\(\chi_{5610}(2557,\cdot)\)
\(\chi_{5610}(2623,\cdot)\)
\(\chi_{5610}(2953,\cdot)\)
\(\chi_{5610}(3067,\cdot)\)
\(\chi_{5610}(3133,\cdot)\)
\(\chi_{5610}(3193,\cdot)\)
\(\chi_{5610}(3403,\cdot)\)
\(\chi_{5610}(3457,\cdot)\)
\(\chi_{5610}(3463,\cdot)\)
\(\chi_{5610}(3547,\cdot)\)
\(\chi_{5610}(4057,\cdot)\)
\(\chi_{5610}(4447,\cdot)\)
\(\chi_{5610}(4723,\cdot)\)
\(\chi_{5610}(4933,\cdot)\)
\(\chi_{5610}(4987,\cdot)\)
\(\chi_{5610}(5107,\cdot)\)
\(\chi_{5610}(5173,\cdot)\)
...
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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 ]
\((1871,3367,1531,3301)\) → \((1,-i,e\left(\frac{4}{5}\right),e\left(\frac{1}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 5610 }(1873, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) |
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sage:chi.jacobi_sum(n)
