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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15730, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,4,11]))
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pari:[g,chi] = znchar(Mod(3323,15730))
\(\chi_{15730}(463,\cdot)\)
\(\chi_{15730}(837,\cdot)\)
\(\chi_{15730}(1893,\cdot)\)
\(\chi_{15730}(2267,\cdot)\)
\(\chi_{15730}(3323,\cdot)\)
\(\chi_{15730}(3697,\cdot)\)
\(\chi_{15730}(4753,\cdot)\)
\(\chi_{15730}(5127,\cdot)\)
\(\chi_{15730}(6183,\cdot)\)
\(\chi_{15730}(6557,\cdot)\)
\(\chi_{15730}(7613,\cdot)\)
\(\chi_{15730}(9043,\cdot)\)
\(\chi_{15730}(9417,\cdot)\)
\(\chi_{15730}(10473,\cdot)\)
\(\chi_{15730}(10847,\cdot)\)
\(\chi_{15730}(11903,\cdot)\)
\(\chi_{15730}(12277,\cdot)\)
\(\chi_{15730}(13333,\cdot)\)
\(\chi_{15730}(13707,\cdot)\)
\(\chi_{15730}(15137,\cdot)\)
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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 ]
\((3147,3511,1211)\) → \((-i,e\left(\frac{1}{11}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 15730 }(3323, a) \) |
\(1\) | \(1\) | \(i\) | \(e\left(\frac{3}{22}\right)\) | \(-1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(-i\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) |
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sage:chi.jacobi_sum(n)
