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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15730, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([15,6,10]))
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pari:[g,chi] = znchar(Mod(5563,15730))
\(\chi_{15730}(233,\cdot)\)
\(\chi_{15730}(2417,\cdot)\)
\(\chi_{15730}(5563,\cdot)\)
\(\chi_{15730}(11777,\cdot)\)
\(\chi_{15730}(12557,\cdot)\)
\(\chi_{15730}(12817,\cdot)\)
\(\chi_{15730}(14923,\cdot)\)
\(\chi_{15730}(15703,\cdot)\)
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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 ]
\((3147,3511,1211)\) → \((-i,e\left(\frac{3}{10}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 15730 }(5563, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(1\) | \(i\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) |
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sage:chi.jacobi_sum(n)
