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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4641, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,16,28,21]))
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pari:[g,chi] = znchar(Mod(2221,4641))
\(\chi_{4641}(184,\cdot)\)
\(\chi_{4641}(760,\cdot)\)
\(\chi_{4641}(856,\cdot)\)
\(\chi_{4641}(1129,\cdot)\)
\(\chi_{4641}(1306,\cdot)\)
\(\chi_{4641}(1822,\cdot)\)
\(\chi_{4641}(1948,\cdot)\)
\(\chi_{4641}(2221,\cdot)\)
\(\chi_{4641}(2368,\cdot)\)
\(\chi_{4641}(2944,\cdot)\)
\(\chi_{4641}(3049,\cdot)\)
\(\chi_{4641}(3322,\cdot)\)
\(\chi_{4641}(3490,\cdot)\)
\(\chi_{4641}(4141,\cdot)\)
\(\chi_{4641}(4279,\cdot)\)
\(\chi_{4641}(4414,\cdot)\)
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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 ]
\((3095,3979,3928,547)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{7}{12}\right),e\left(\frac{7}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(19\) | \(20\) | \(22\) |
| \( \chi_{ 4641 }(2221, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(-1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) |
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sage:chi.jacobi_sum(n)
