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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15730, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,24,20]))
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pari:[g,chi] = znchar(Mod(10409,15730))
\(\chi_{15730}(9,\cdot)\)
\(\chi_{15730}(269,\cdot)\)
\(\chi_{15730}(1049,\cdot)\)
\(\chi_{15730}(3149,\cdot)\)
\(\chi_{15730}(8479,\cdot)\)
\(\chi_{15730}(8739,\cdot)\)
\(\chi_{15730}(9519,\cdot)\)
\(\chi_{15730}(10409,\cdot)\)
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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 ]
\((3147,3511,1211)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 15730 }(10409, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) |
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sage:chi.jacobi_sum(n)
