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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5712, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,36,0,32,45]))
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pari:[g,chi] = znchar(Mod(4477,5712))
\(\chi_{5712}(37,\cdot)\)
\(\chi_{5712}(109,\cdot)\)
\(\chi_{5712}(277,\cdot)\)
\(\chi_{5712}(709,\cdot)\)
\(\chi_{5712}(877,\cdot)\)
\(\chi_{5712}(949,\cdot)\)
\(\chi_{5712}(1213,\cdot)\)
\(\chi_{5712}(2221,\cdot)\)
\(\chi_{5712}(2557,\cdot)\)
\(\chi_{5712}(2725,\cdot)\)
\(\chi_{5712}(3301,\cdot)\)
\(\chi_{5712}(3397,\cdot)\)
\(\chi_{5712}(3973,\cdot)\)
\(\chi_{5712}(4141,\cdot)\)
\(\chi_{5712}(4477,\cdot)\)
\(\chi_{5712}(5485,\cdot)\)
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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 ]
\((2143,1429,3809,3265,2689)\) → \((1,-i,1,e\left(\frac{2}{3}\right),e\left(\frac{15}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5712 }(4477, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) |
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sage:chi.jacobi_sum(n)
