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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7448, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([63,0,99,56]))
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pari:[g,chi] = znchar(Mod(503,7448))
\(\chi_{7448}(55,\cdot)\)
\(\chi_{7448}(111,\cdot)\)
\(\chi_{7448}(503,\cdot)\)
\(\chi_{7448}(671,\cdot)\)
\(\chi_{7448}(727,\cdot)\)
\(\chi_{7448}(1119,\cdot)\)
\(\chi_{7448}(1735,\cdot)\)
\(\chi_{7448}(1791,\cdot)\)
\(\chi_{7448}(1847,\cdot)\)
\(\chi_{7448}(2183,\cdot)\)
\(\chi_{7448}(2239,\cdot)\)
\(\chi_{7448}(2631,\cdot)\)
\(\chi_{7448}(2799,\cdot)\)
\(\chi_{7448}(2855,\cdot)\)
\(\chi_{7448}(2911,\cdot)\)
\(\chi_{7448}(3247,\cdot)\)
\(\chi_{7448}(3303,\cdot)\)
\(\chi_{7448}(3695,\cdot)\)
\(\chi_{7448}(3863,\cdot)\)
\(\chi_{7448}(3975,\cdot)\)
\(\chi_{7448}(4367,\cdot)\)
\(\chi_{7448}(4759,\cdot)\)
\(\chi_{7448}(4927,\cdot)\)
\(\chi_{7448}(4983,\cdot)\)
\(\chi_{7448}(5039,\cdot)\)
\(\chi_{7448}(5375,\cdot)\)
\(\chi_{7448}(5431,\cdot)\)
\(\chi_{7448}(5823,\cdot)\)
\(\chi_{7448}(5991,\cdot)\)
\(\chi_{7448}(6047,\cdot)\)
...
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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 ]
\((1863,3725,3041,3137)\) → \((-1,1,e\left(\frac{11}{14}\right),e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 7448 }(503, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) |
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sage:chi.jacobi_sum(n)
