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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([63,0,57,56]))
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pari:[g,chi] = znchar(Mod(1263,7448))
\(\chi_{7448}(199,\cdot)\)
\(\chi_{7448}(271,\cdot)\)
\(\chi_{7448}(367,\cdot)\)
\(\chi_{7448}(479,\cdot)\)
\(\chi_{7448}(663,\cdot)\)
\(\chi_{7448}(719,\cdot)\)
\(\chi_{7448}(1263,\cdot)\)
\(\chi_{7448}(1335,\cdot)\)
\(\chi_{7448}(1431,\cdot)\)
\(\chi_{7448}(1543,\cdot)\)
\(\chi_{7448}(1727,\cdot)\)
\(\chi_{7448}(2327,\cdot)\)
\(\chi_{7448}(2399,\cdot)\)
\(\chi_{7448}(2495,\cdot)\)
\(\chi_{7448}(2607,\cdot)\)
\(\chi_{7448}(2791,\cdot)\)
\(\chi_{7448}(2847,\cdot)\)
\(\chi_{7448}(3391,\cdot)\)
\(\chi_{7448}(3463,\cdot)\)
\(\chi_{7448}(3671,\cdot)\)
\(\chi_{7448}(3855,\cdot)\)
\(\chi_{7448}(3911,\cdot)\)
\(\chi_{7448}(4455,\cdot)\)
\(\chi_{7448}(4623,\cdot)\)
\(\chi_{7448}(4975,\cdot)\)
\(\chi_{7448}(5591,\cdot)\)
\(\chi_{7448}(5687,\cdot)\)
\(\chi_{7448}(5799,\cdot)\)
\(\chi_{7448}(5983,\cdot)\)
\(\chi_{7448}(6039,\cdot)\)
...
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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 ]
\((1863,3725,3041,3137)\) → \((-1,1,e\left(\frac{19}{42}\right),e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 7448 }(1263, a) \) |
\(1\) | \(1\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) |
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sage:chi.jacobi_sum(n)
