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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4725, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,3,50]))
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pari:[g,chi] = znchar(Mod(1027,4725))
\(\chi_{4725}(703,\cdot)\)
\(\chi_{4725}(838,\cdot)\)
\(\chi_{4725}(892,\cdot)\)
\(\chi_{4725}(1027,\cdot)\)
\(\chi_{4725}(1648,\cdot)\)
\(\chi_{4725}(1783,\cdot)\)
\(\chi_{4725}(1837,\cdot)\)
\(\chi_{4725}(1972,\cdot)\)
\(\chi_{4725}(2728,\cdot)\)
\(\chi_{4725}(2917,\cdot)\)
\(\chi_{4725}(3538,\cdot)\)
\(\chi_{4725}(3673,\cdot)\)
\(\chi_{4725}(3727,\cdot)\)
\(\chi_{4725}(3862,\cdot)\)
\(\chi_{4725}(4483,\cdot)\)
\(\chi_{4725}(4672,\cdot)\)
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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 ]
\((4376,1702,2026)\) → \((1,e\left(\frac{1}{20}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 4725 }(1027, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) |
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sage:chi.jacobi_sum(n)
