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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([0,24,8,16,21]))
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pari:[g,chi] = znchar(Mod(13169,34272))
\(\chi_{34272}(3089,\cdot)\)
\(\chi_{34272}(5009,\cdot)\)
\(\chi_{34272}(5105,\cdot)\)
\(\chi_{34272}(9041,\cdot)\)
\(\chi_{34272}(11057,\cdot)\)
\(\chi_{34272}(13169,\cdot)\)
\(\chi_{34272}(15089,\cdot)\)
\(\chi_{34272}(17105,\cdot)\)
\(\chi_{34272}(17201,\cdot)\)
\(\chi_{34272}(19217,\cdot)\)
\(\chi_{34272}(21137,\cdot)\)
\(\chi_{34272}(23249,\cdot)\)
\(\chi_{34272}(25265,\cdot)\)
\(\chi_{34272}(29201,\cdot)\)
\(\chi_{34272}(29297,\cdot)\)
\(\chi_{34272}(31217,\cdot)\)
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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 ]
\((2143,29989,3809,14689,14113)\) → \((1,-1,e\left(\frac{1}{6}\right),e\left(\frac{1}{3}\right),e\left(\frac{7}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 34272 }(13169, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) |
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sage:chi.jacobi_sum(n)
