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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5200, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,42,55]))
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pari:[g,chi] = znchar(Mod(1359,5200))
\(\chi_{5200}(319,\cdot)\)
\(\chi_{5200}(479,\cdot)\)
\(\chi_{5200}(639,\cdot)\)
\(\chi_{5200}(1359,\cdot)\)
\(\chi_{5200}(1519,\cdot)\)
\(\chi_{5200}(1679,\cdot)\)
\(\chi_{5200}(1839,\cdot)\)
\(\chi_{5200}(2559,\cdot)\)
\(\chi_{5200}(2719,\cdot)\)
\(\chi_{5200}(2879,\cdot)\)
\(\chi_{5200}(3439,\cdot)\)
\(\chi_{5200}(3759,\cdot)\)
\(\chi_{5200}(3919,\cdot)\)
\(\chi_{5200}(4479,\cdot)\)
\(\chi_{5200}(4639,\cdot)\)
\(\chi_{5200}(4959,\cdot)\)
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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 ]
\((1951,1301,4577,1601)\) → \((-1,1,e\left(\frac{7}{10}\right),e\left(\frac{11}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 5200 }(1359, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) |
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sage:chi.jacobi_sum(n)
