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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5200, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,12,5]))
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pari:[g,chi] = znchar(Mod(671,5200))
\(\chi_{5200}(31,\cdot)\)
\(\chi_{5200}(671,\cdot)\)
\(\chi_{5200}(1071,\cdot)\)
\(\chi_{5200}(1711,\cdot)\)
\(\chi_{5200}(2111,\cdot)\)
\(\chi_{5200}(3791,\cdot)\)
\(\chi_{5200}(4191,\cdot)\)
\(\chi_{5200}(4831,\cdot)\)
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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 ]
\((1951,1301,4577,1601)\) → \((-1,1,e\left(\frac{3}{5}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 5200 }(671, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) |
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sage:chi.jacobi_sum(n)
