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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([45,42,10]))
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pari:[g,chi] = znchar(Mod(5243,15730))
\(\chi_{15730}(2097,\cdot)\)
\(\chi_{15730}(2877,\cdot)\)
\(\chi_{15730}(3137,\cdot)\)
\(\chi_{15730}(3143,\cdot)\)
\(\chi_{15730}(5243,\cdot)\)
\(\chi_{15730}(6023,\cdot)\)
\(\chi_{15730}(6283,\cdot)\)
\(\chi_{15730}(8467,\cdot)\)
\(\chi_{15730}(9357,\cdot)\)
\(\chi_{15730}(10137,\cdot)\)
\(\chi_{15730}(10397,\cdot)\)
\(\chi_{15730}(11613,\cdot)\)
\(\chi_{15730}(12503,\cdot)\)
\(\chi_{15730}(13283,\cdot)\)
\(\chi_{15730}(13543,\cdot)\)
\(\chi_{15730}(15727,\cdot)\)
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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 ]
\((3147,3511,1211)\) → \((-i,e\left(\frac{7}{10}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 15730 }(5243, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) |
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sage:chi.jacobi_sum(n)
