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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(34272, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,0,8,0,9]))
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pari:[g,chi] = znchar(Mod(15905,34272))
\(\chi_{34272}(5825,\cdot)\)
\(\chi_{34272}(7169,\cdot)\)
\(\chi_{34272}(9185,\cdot)\)
\(\chi_{34272}(9857,\cdot)\)
\(\chi_{34272}(11873,\cdot)\)
\(\chi_{34272}(15905,\cdot)\)
\(\chi_{34272}(17249,\cdot)\)
\(\chi_{34272}(17921,\cdot)\)
\(\chi_{34272}(21281,\cdot)\)
\(\chi_{34272}(21953,\cdot)\)
\(\chi_{34272}(23297,\cdot)\)
\(\chi_{34272}(27329,\cdot)\)
\(\chi_{34272}(29345,\cdot)\)
\(\chi_{34272}(30017,\cdot)\)
\(\chi_{34272}(32033,\cdot)\)
\(\chi_{34272}(33377,\cdot)\)
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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 ]
\((2143,29989,3809,14689,14113)\) → \((1,1,e\left(\frac{1}{6}\right),1,e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 34272 }(15905, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{43}{48}\right)\) |
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sage:chi.jacobi_sum(n)
