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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5712, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,8,8,0,11]))
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pari:[g,chi] = znchar(Mod(5447,5712))
\(\chi_{5712}(71,\cdot)\)
\(\chi_{5712}(743,\cdot)\)
\(\chi_{5712}(2759,\cdot)\)
\(\chi_{5712}(3431,\cdot)\)
\(\chi_{5712}(3767,\cdot)\)
\(\chi_{5712}(4103,\cdot)\)
\(\chi_{5712}(5111,\cdot)\)
\(\chi_{5712}(5447,\cdot)\)
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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 ]
\((2143,1429,3809,3265,2689)\) → \((-1,-1,-1,1,e\left(\frac{11}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5712 }(5447, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) |
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sage:chi.jacobi_sum(n)
