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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2448, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,12,16,15]))
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pari:[g,chi] = znchar(Mod(25,2448))
\(\chi_{2448}(25,\cdot)\)
\(\chi_{2448}(121,\cdot)\)
\(\chi_{2448}(457,\cdot)\)
\(\chi_{2448}(553,\cdot)\)
\(\chi_{2448}(841,\cdot)\)
\(\chi_{2448}(1273,\cdot)\)
\(\chi_{2448}(1753,\cdot)\)
\(\chi_{2448}(2185,\cdot)\)
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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 ]
\((2143,613,1361,1873)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{5}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2448 }(25, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
