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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(508288, base_ring=CyclotomicField(13680))
M = H._module
chi = DirichletCharacter(H, M([0,2565,8208,11120]))
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pari:[g,chi] = znchar(Mod(9,508288))
\(\chi_{508288}(9,\cdot)\)
\(\chi_{508288}(25,\cdot)\)
\(\chi_{508288}(137,\cdot)\)
\(\chi_{508288}(169,\cdot)\)
\(\chi_{508288}(313,\cdot)\)
\(\chi_{508288}(377,\cdot)\)
\(\chi_{508288}(537,\cdot)\)
\(\chi_{508288}(841,\cdot)\)
\(\chi_{508288}(1049,\cdot)\)
\(\chi_{508288}(1081,\cdot)\)
\(\chi_{508288}(1225,\cdot)\)
\(\chi_{508288}(1241,\cdot)\)
\(\chi_{508288}(1545,\cdot)\)
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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 ]
\((166783,174725,323457,14081)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{3}{5}\right),e\left(\frac{139}{171}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 508288 }(9, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4799}{13680}\right)\) | \(e\left(\frac{2357}{13680}\right)\) | \(e\left(\frac{11}{2280}\right)\) | \(e\left(\frac{4799}{6840}\right)\) | \(e\left(\frac{11563}{13680}\right)\) | \(e\left(\frac{1789}{3420}\right)\) | \(e\left(\frac{1223}{3420}\right)\) | \(e\left(\frac{973}{2736}\right)\) | \(e\left(\frac{415}{1368}\right)\) | \(e\left(\frac{2357}{6840}\right)\) |
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sage:chi.jacobi_sum(n)
