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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([112,90]))
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pari:[g,chi] = znchar(Mod(748,2349))
\(\chi_{2349}(181,\cdot)\)
\(\chi_{2349}(199,\cdot)\)
\(\chi_{2349}(226,\cdot)\)
\(\chi_{2349}(343,\cdot)\)
\(\chi_{2349}(397,\cdot)\)
\(\chi_{2349}(442,\cdot)\)
\(\chi_{2349}(451,\cdot)\)
\(\chi_{2349}(604,\cdot)\)
\(\chi_{2349}(658,\cdot)\)
\(\chi_{2349}(712,\cdot)\)
\(\chi_{2349}(721,\cdot)\)
\(\chi_{2349}(748,\cdot)\)
\(\chi_{2349}(964,\cdot)\)
\(\chi_{2349}(982,\cdot)\)
\(\chi_{2349}(1009,\cdot)\)
\(\chi_{2349}(1126,\cdot)\)
\(\chi_{2349}(1180,\cdot)\)
\(\chi_{2349}(1225,\cdot)\)
\(\chi_{2349}(1234,\cdot)\)
\(\chi_{2349}(1387,\cdot)\)
\(\chi_{2349}(1441,\cdot)\)
\(\chi_{2349}(1495,\cdot)\)
\(\chi_{2349}(1504,\cdot)\)
\(\chi_{2349}(1531,\cdot)\)
\(\chi_{2349}(1747,\cdot)\)
\(\chi_{2349}(1765,\cdot)\)
\(\chi_{2349}(1792,\cdot)\)
\(\chi_{2349}(1909,\cdot)\)
\(\chi_{2349}(1963,\cdot)\)
\(\chi_{2349}(2008,\cdot)\)
...
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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 ]
\((407,1945)\) → \((e\left(\frac{8}{9}\right),e\left(\frac{5}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(748, a) \) |
\(1\) | \(1\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{26}{63}\right)\) |
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sage:chi.jacobi_sum(n)
