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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7448, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,15,7]))
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pari:[g,chi] = znchar(Mod(825,7448))
\(\chi_{7448}(601,\cdot)\)
\(\chi_{7448}(825,\cdot)\)
\(\chi_{7448}(1889,\cdot)\)
\(\chi_{7448}(2729,\cdot)\)
\(\chi_{7448}(2953,\cdot)\)
\(\chi_{7448}(3793,\cdot)\)
\(\chi_{7448}(4857,\cdot)\)
\(\chi_{7448}(5081,\cdot)\)
\(\chi_{7448}(5921,\cdot)\)
\(\chi_{7448}(6145,\cdot)\)
\(\chi_{7448}(6985,\cdot)\)
\(\chi_{7448}(7209,\cdot)\)
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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 ]
\((1863,3725,3041,3137)\) → \((1,1,e\left(\frac{5}{14}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 7448 }(825, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) |
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sage:chi.jacobi_sum(n)
