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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2349, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([7,36]))
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pari:[g,chi] = znchar(Mod(575,2349))
\(\chi_{2349}(152,\cdot)\)
\(\chi_{2349}(170,\cdot)\)
\(\chi_{2349}(197,\cdot)\)
\(\chi_{2349}(314,\cdot)\)
\(\chi_{2349}(368,\cdot)\)
\(\chi_{2349}(413,\cdot)\)
\(\chi_{2349}(422,\cdot)\)
\(\chi_{2349}(575,\cdot)\)
\(\chi_{2349}(629,\cdot)\)
\(\chi_{2349}(683,\cdot)\)
\(\chi_{2349}(692,\cdot)\)
\(\chi_{2349}(719,\cdot)\)
\(\chi_{2349}(935,\cdot)\)
\(\chi_{2349}(953,\cdot)\)
\(\chi_{2349}(980,\cdot)\)
\(\chi_{2349}(1097,\cdot)\)
\(\chi_{2349}(1151,\cdot)\)
\(\chi_{2349}(1196,\cdot)\)
\(\chi_{2349}(1205,\cdot)\)
\(\chi_{2349}(1358,\cdot)\)
\(\chi_{2349}(1412,\cdot)\)
\(\chi_{2349}(1466,\cdot)\)
\(\chi_{2349}(1475,\cdot)\)
\(\chi_{2349}(1502,\cdot)\)
\(\chi_{2349}(1718,\cdot)\)
\(\chi_{2349}(1736,\cdot)\)
\(\chi_{2349}(1763,\cdot)\)
\(\chi_{2349}(1880,\cdot)\)
\(\chi_{2349}(1934,\cdot)\)
\(\chi_{2349}(1979,\cdot)\)
...
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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 ]
\((407,1945)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{2}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(575, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{23}{63}\right)\) |
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sage:chi.jacobi_sum(n)
