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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2349, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([14,57]))
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pari:[g,chi] = znchar(Mod(26,2349))
\(\chi_{2349}(26,\cdot)\)
\(\chi_{2349}(134,\cdot)\)
\(\chi_{2349}(188,\cdot)\)
\(\chi_{2349}(269,\cdot)\)
\(\chi_{2349}(350,\cdot)\)
\(\chi_{2349}(512,\cdot)\)
\(\chi_{2349}(620,\cdot)\)
\(\chi_{2349}(917,\cdot)\)
\(\chi_{2349}(1025,\cdot)\)
\(\chi_{2349}(1187,\cdot)\)
\(\chi_{2349}(1268,\cdot)\)
\(\chi_{2349}(1349,\cdot)\)
\(\chi_{2349}(1403,\cdot)\)
\(\chi_{2349}(1511,\cdot)\)
\(\chi_{2349}(1592,\cdot)\)
\(\chi_{2349}(1754,\cdot)\)
\(\chi_{2349}(1808,\cdot)\)
\(\chi_{2349}(1835,\cdot)\)
\(\chi_{2349}(1916,\cdot)\)
\(\chi_{2349}(1970,\cdot)\)
\(\chi_{2349}(2051,\cdot)\)
\(\chi_{2349}(2078,\cdot)\)
\(\chi_{2349}(2132,\cdot)\)
\(\chi_{2349}(2294,\cdot)\)
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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 ]
\((407,1945)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{19}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(26, a) \) |
\(1\) | \(1\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{8}{21}\right)\) |
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sage:chi.jacobi_sum(n)
