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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40320, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,21,0,0,16]))
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pari:[g,chi] = znchar(Mod(10891,40320))
\(\chi_{40320}(811,\cdot)\)
\(\chi_{40320}(3331,\cdot)\)
\(\chi_{40320}(5851,\cdot)\)
\(\chi_{40320}(8371,\cdot)\)
\(\chi_{40320}(10891,\cdot)\)
\(\chi_{40320}(13411,\cdot)\)
\(\chi_{40320}(15931,\cdot)\)
\(\chi_{40320}(18451,\cdot)\)
\(\chi_{40320}(20971,\cdot)\)
\(\chi_{40320}(23491,\cdot)\)
\(\chi_{40320}(26011,\cdot)\)
\(\chi_{40320}(28531,\cdot)\)
\(\chi_{40320}(31051,\cdot)\)
\(\chi_{40320}(33571,\cdot)\)
\(\chi_{40320}(36091,\cdot)\)
\(\chi_{40320}(38611,\cdot)\)
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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 ]
\((8191,23941,17921,32257,28801)\) → \((-1,e\left(\frac{21}{32}\right),1,1,-1)\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 40320 }(10891, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(i\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{17}{32}\right)\) |
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sage:chi.jacobi_sum(n)
