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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15730, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,36,5]))
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pari:[g,chi] = znchar(Mod(977,15730))
\(\chi_{15730}(323,\cdot)\)
\(\chi_{15730}(487,\cdot)\)
\(\chi_{15730}(977,\cdot)\)
\(\chi_{15730}(1237,\cdot)\)
\(\chi_{15730}(2017,\cdot)\)
\(\chi_{15730}(4843,\cdot)\)
\(\chi_{15730}(5817,\cdot)\)
\(\chi_{15730}(6077,\cdot)\)
\(\chi_{15730}(6857,\cdot)\)
\(\chi_{15730}(9683,\cdot)\)
\(\chi_{15730}(10173,\cdot)\)
\(\chi_{15730}(10433,\cdot)\)
\(\chi_{15730}(11213,\cdot)\)
\(\chi_{15730}(11377,\cdot)\)
\(\chi_{15730}(15013,\cdot)\)
\(\chi_{15730}(15273,\cdot)\)
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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 ]
\((3147,3511,1211)\) → \((i,e\left(\frac{3}{5}\right),e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 15730 }(977, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) |
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sage:chi.jacobi_sum(n)
