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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25410, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,10,24]))
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pari:[g,chi] = znchar(Mod(21541,25410))
\(\chi_{25410}(2671,\cdot)\)
\(\chi_{25410}(4141,\cdot)\)
\(\chi_{25410}(8551,\cdot)\)
\(\chi_{25410}(10651,\cdot)\)
\(\chi_{25410}(13561,\cdot)\)
\(\chi_{25410}(15031,\cdot)\)
\(\chi_{25410}(19441,\cdot)\)
\(\chi_{25410}(21541,\cdot)\)
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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 ]
\((8471,15247,14521,7141)\) → \((1,1,e\left(\frac{1}{3}\right),e\left(\frac{4}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 25410 }(21541, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(e\left(\frac{1}{15}\right)\) |
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sage:chi.jacobi_sum(n)
