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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25410, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,0,42]))
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pari:[g,chi] = znchar(Mod(19447,25410))
\(\chi_{25410}(43,\cdot)\)
\(\chi_{25410}(2353,\cdot)\)
\(\chi_{25410}(3277,\cdot)\)
\(\chi_{25410}(4663,\cdot)\)
\(\chi_{25410}(5587,\cdot)\)
\(\chi_{25410}(6973,\cdot)\)
\(\chi_{25410}(7897,\cdot)\)
\(\chi_{25410}(9283,\cdot)\)
\(\chi_{25410}(10207,\cdot)\)
\(\chi_{25410}(11593,\cdot)\)
\(\chi_{25410}(12517,\cdot)\)
\(\chi_{25410}(13903,\cdot)\)
\(\chi_{25410}(14827,\cdot)\)
\(\chi_{25410}(17137,\cdot)\)
\(\chi_{25410}(18523,\cdot)\)
\(\chi_{25410}(19447,\cdot)\)
\(\chi_{25410}(20833,\cdot)\)
\(\chi_{25410}(21757,\cdot)\)
\(\chi_{25410}(23143,\cdot)\)
\(\chi_{25410}(24067,\cdot)\)
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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 ]
\((8471,15247,14521,7141)\) → \((1,i,1,e\left(\frac{21}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 25410 }(19447, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) |
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sage:chi.jacobi_sum(n)
