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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,17,8]))
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pari:[g,chi] = znchar(Mod(2447,2475))
\(\chi_{2475}(323,\cdot)\)
\(\chi_{2475}(1142,\cdot)\)
\(\chi_{2475}(1763,\cdot)\)
\(\chi_{2475}(1853,\cdot)\)
\(\chi_{2475}(1862,\cdot)\)
\(\chi_{2475}(2033,\cdot)\)
\(\chi_{2475}(2402,\cdot)\)
\(\chi_{2475}(2447,\cdot)\)
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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 ]
\((551,2377,2026)\) → \((-1,e\left(\frac{17}{20}\right),e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 2475 }(2447, a) \) |
\(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{1}{20}\right)\) | \(i\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(-1\) | \(e\left(\frac{17}{20}\right)\) |
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sage:chi.jacobi_sum(n)
