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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([24,0,8,27]))
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pari:[g,chi] = znchar(Mod(4502,4641))
\(\chi_{4641}(218,\cdot)\)
\(\chi_{4641}(1499,\cdot)\)
\(\chi_{4641}(1856,\cdot)\)
\(\chi_{4641}(2045,\cdot)\)
\(\chi_{4641}(2318,\cdot)\)
\(\chi_{4641}(2402,\cdot)\)
\(\chi_{4641}(2591,\cdot)\)
\(\chi_{4641}(2675,\cdot)\)
\(\chi_{4641}(2948,\cdot)\)
\(\chi_{4641}(3410,\cdot)\)
\(\chi_{4641}(3683,\cdot)\)
\(\chi_{4641}(3767,\cdot)\)
\(\chi_{4641}(3956,\cdot)\)
\(\chi_{4641}(4040,\cdot)\)
\(\chi_{4641}(4313,\cdot)\)
\(\chi_{4641}(4502,\cdot)\)
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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 ]
\((3095,3979,3928,547)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{9}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(19\) | \(20\) | \(22\) |
| \( \chi_{ 4641 }(4502, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) |
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sage:chi.jacobi_sum(n)
