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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([63,0,102,7]))
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pari:[g,chi] = znchar(Mod(3175,7448))
\(\chi_{7448}(375,\cdot)\)
\(\chi_{7448}(583,\cdot)\)
\(\chi_{7448}(751,\cdot)\)
\(\chi_{7448}(991,\cdot)\)
\(\chi_{7448}(1535,\cdot)\)
\(\chi_{7448}(1815,\cdot)\)
\(\chi_{7448}(2055,\cdot)\)
\(\chi_{7448}(2111,\cdot)\)
\(\chi_{7448}(2503,\cdot)\)
\(\chi_{7448}(2599,\cdot)\)
\(\chi_{7448}(2711,\cdot)\)
\(\chi_{7448}(2879,\cdot)\)
\(\chi_{7448}(3119,\cdot)\)
\(\chi_{7448}(3175,\cdot)\)
\(\chi_{7448}(3567,\cdot)\)
\(\chi_{7448}(3663,\cdot)\)
\(\chi_{7448}(3775,\cdot)\)
\(\chi_{7448}(3943,\cdot)\)
\(\chi_{7448}(4239,\cdot)\)
\(\chi_{7448}(4631,\cdot)\)
\(\chi_{7448}(4727,\cdot)\)
\(\chi_{7448}(4839,\cdot)\)
\(\chi_{7448}(5007,\cdot)\)
\(\chi_{7448}(5247,\cdot)\)
\(\chi_{7448}(5303,\cdot)\)
\(\chi_{7448}(5695,\cdot)\)
\(\chi_{7448}(5791,\cdot)\)
\(\chi_{7448}(5903,\cdot)\)
\(\chi_{7448}(6071,\cdot)\)
\(\chi_{7448}(6311,\cdot)\)
...
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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 ]
\((1863,3725,3041,3137)\) → \((-1,1,e\left(\frac{17}{21}\right),e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 7448 }(3175, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) |
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sage:chi.jacobi_sum(n)
