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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3025, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([3,2]))
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pari:[g,chi] = znchar(Mod(233,3025))
\(\chi_{3025}(233,\cdot)\)
\(\chi_{3025}(578,\cdot)\)
\(\chi_{3025}(717,\cdot)\)
\(\chi_{3025}(1812,\cdot)\)
\(\chi_{3025}(2272,\cdot)\)
\(\chi_{3025}(2538,\cdot)\)
\(\chi_{3025}(2702,\cdot)\)
\(\chi_{3025}(2823,\cdot)\)
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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 ]
\((727,2301)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{1}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 3025 }(233, a) \) |
\(1\) | \(1\) | \(i\) | \(e\left(\frac{17}{20}\right)\) | \(-1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(-i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) |
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sage:chi.jacobi_sum(n)
