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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9984, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,13,0,0]))
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pari:[g,chi] = znchar(Mod(79,9984))
\(\chi_{9984}(79,\cdot)\)
\(\chi_{9984}(1327,\cdot)\)
\(\chi_{9984}(2575,\cdot)\)
\(\chi_{9984}(3823,\cdot)\)
\(\chi_{9984}(5071,\cdot)\)
\(\chi_{9984}(6319,\cdot)\)
\(\chi_{9984}(7567,\cdot)\)
\(\chi_{9984}(8815,\cdot)\)
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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 ]
\((8191,3589,3329,769)\) → \((-1,e\left(\frac{13}{16}\right),1,1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 9984 }(79, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(1\) | \(e\left(\frac{7}{16}\right)\) |
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sage:chi.jacobi_sum(n)
