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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1576, base_ring=CyclotomicField(98))
M = H._module
chi = DirichletCharacter(H, M([0,0,48]))
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pari:[g,chi] = znchar(Mod(49,1576))
\(\chi_{1576}(49,\cdot)\)
\(\chi_{1576}(81,\cdot)\)
\(\chi_{1576}(105,\cdot)\)
\(\chi_{1576}(193,\cdot)\)
\(\chi_{1576}(225,\cdot)\)
\(\chi_{1576}(257,\cdot)\)
\(\chi_{1576}(273,\cdot)\)
\(\chi_{1576}(297,\cdot)\)
\(\chi_{1576}(329,\cdot)\)
\(\chi_{1576}(353,\cdot)\)
\(\chi_{1576}(369,\cdot)\)
\(\chi_{1576}(385,\cdot)\)
\(\chi_{1576}(417,\cdot)\)
\(\chi_{1576}(457,\cdot)\)
\(\chi_{1576}(529,\cdot)\)
\(\chi_{1576}(569,\cdot)\)
\(\chi_{1576}(625,\cdot)\)
\(\chi_{1576}(633,\cdot)\)
\(\chi_{1576}(681,\cdot)\)
\(\chi_{1576}(745,\cdot)\)
\(\chi_{1576}(817,\cdot)\)
\(\chi_{1576}(825,\cdot)\)
\(\chi_{1576}(841,\cdot)\)
\(\chi_{1576}(849,\cdot)\)
\(\chi_{1576}(873,\cdot)\)
\(\chi_{1576}(889,\cdot)\)
\(\chi_{1576}(921,\cdot)\)
\(\chi_{1576}(1001,\cdot)\)
\(\chi_{1576}(1009,\cdot)\)
\(\chi_{1576}(1025,\cdot)\)
...
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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 ]
\((1183,789,593)\) → \((1,1,e\left(\frac{24}{49}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1576 }(49, a) \) |
\(1\) | \(1\) | \(e\left(\frac{32}{49}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{25}{49}\right)\) | \(e\left(\frac{15}{49}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{43}{49}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{8}{49}\right)\) |
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sage:chi.jacobi_sum(n)
