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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([24,42,24,40,21]))
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pari:[g,chi] = znchar(Mod(4499,34272))
\(\chi_{34272}(2987,\cdot)\)
\(\chi_{34272}(4499,\cdot)\)
\(\chi_{34272}(5003,\cdot)\)
\(\chi_{34272}(5939,\cdot)\)
\(\chi_{34272}(8027,\cdot)\)
\(\chi_{34272}(12995,\cdot)\)
\(\chi_{34272}(17027,\cdot)\)
\(\chi_{34272}(19547,\cdot)\)
\(\chi_{34272}(20627,\cdot)\)
\(\chi_{34272}(22571,\cdot)\)
\(\chi_{34272}(24083,\cdot)\)
\(\chi_{34272}(24587,\cdot)\)
\(\chi_{34272}(27611,\cdot)\)
\(\chi_{34272}(27683,\cdot)\)
\(\chi_{34272}(31715,\cdot)\)
\(\chi_{34272}(34235,\cdot)\)
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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 ]
\((2143,29989,3809,14689,14113)\) → \((-1,e\left(\frac{7}{8}\right),-1,e\left(\frac{5}{6}\right),e\left(\frac{7}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 34272 }(4499, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{16}\right)\) |
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sage:chi.jacobi_sum(n)
