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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4864, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([48,15,32]))
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pari:[g,chi] = znchar(Mod(7,4864))
\(\chi_{4864}(7,\cdot)\)
\(\chi_{4864}(87,\cdot)\)
\(\chi_{4864}(311,\cdot)\)
\(\chi_{4864}(391,\cdot)\)
\(\chi_{4864}(615,\cdot)\)
\(\chi_{4864}(695,\cdot)\)
\(\chi_{4864}(919,\cdot)\)
\(\chi_{4864}(999,\cdot)\)
\(\chi_{4864}(1223,\cdot)\)
\(\chi_{4864}(1303,\cdot)\)
\(\chi_{4864}(1527,\cdot)\)
\(\chi_{4864}(1607,\cdot)\)
\(\chi_{4864}(1831,\cdot)\)
\(\chi_{4864}(1911,\cdot)\)
\(\chi_{4864}(2135,\cdot)\)
\(\chi_{4864}(2215,\cdot)\)
\(\chi_{4864}(2439,\cdot)\)
\(\chi_{4864}(2519,\cdot)\)
\(\chi_{4864}(2743,\cdot)\)
\(\chi_{4864}(2823,\cdot)\)
\(\chi_{4864}(3047,\cdot)\)
\(\chi_{4864}(3127,\cdot)\)
\(\chi_{4864}(3351,\cdot)\)
\(\chi_{4864}(3431,\cdot)\)
\(\chi_{4864}(3655,\cdot)\)
\(\chi_{4864}(3735,\cdot)\)
\(\chi_{4864}(3959,\cdot)\)
\(\chi_{4864}(4039,\cdot)\)
\(\chi_{4864}(4263,\cdot)\)
\(\chi_{4864}(4343,\cdot)\)
...
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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 ]
\((3839,2053,4353)\) → \((-1,e\left(\frac{5}{32}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 4864 }(7, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{35}{96}\right)\) | \(e\left(\frac{17}{48}\right)\) |
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sage:chi.jacobi_sum(n)
