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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9984, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,9,24,16]))
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pari:[g,chi] = znchar(Mod(1199,9984))
\(\chi_{9984}(815,\cdot)\)
\(\chi_{9984}(1199,\cdot)\)
\(\chi_{9984}(2063,\cdot)\)
\(\chi_{9984}(2447,\cdot)\)
\(\chi_{9984}(3311,\cdot)\)
\(\chi_{9984}(3695,\cdot)\)
\(\chi_{9984}(4559,\cdot)\)
\(\chi_{9984}(4943,\cdot)\)
\(\chi_{9984}(5807,\cdot)\)
\(\chi_{9984}(6191,\cdot)\)
\(\chi_{9984}(7055,\cdot)\)
\(\chi_{9984}(7439,\cdot)\)
\(\chi_{9984}(8303,\cdot)\)
\(\chi_{9984}(8687,\cdot)\)
\(\chi_{9984}(9551,\cdot)\)
\(\chi_{9984}(9935,\cdot)\)
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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 ]
\((8191,3589,3329,769)\) → \((-1,e\left(\frac{3}{16}\right),-1,e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 9984 }(1199, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(1\) | \(e\left(\frac{35}{48}\right)\) |
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sage:chi.jacobi_sum(n)
