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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40320, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([48,15,48,24,16]))
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pari:[g,chi] = znchar(Mod(4427,40320))
\(\chi_{40320}(2483,\cdot)\)
\(\chi_{40320}(2987,\cdot)\)
\(\chi_{40320}(3923,\cdot)\)
\(\chi_{40320}(4427,\cdot)\)
\(\chi_{40320}(7523,\cdot)\)
\(\chi_{40320}(8027,\cdot)\)
\(\chi_{40320}(8963,\cdot)\)
\(\chi_{40320}(9467,\cdot)\)
\(\chi_{40320}(12563,\cdot)\)
\(\chi_{40320}(13067,\cdot)\)
\(\chi_{40320}(14003,\cdot)\)
\(\chi_{40320}(14507,\cdot)\)
\(\chi_{40320}(17603,\cdot)\)
\(\chi_{40320}(18107,\cdot)\)
\(\chi_{40320}(19043,\cdot)\)
\(\chi_{40320}(19547,\cdot)\)
\(\chi_{40320}(22643,\cdot)\)
\(\chi_{40320}(23147,\cdot)\)
\(\chi_{40320}(24083,\cdot)\)
\(\chi_{40320}(24587,\cdot)\)
\(\chi_{40320}(27683,\cdot)\)
\(\chi_{40320}(28187,\cdot)\)
\(\chi_{40320}(29123,\cdot)\)
\(\chi_{40320}(29627,\cdot)\)
\(\chi_{40320}(32723,\cdot)\)
\(\chi_{40320}(33227,\cdot)\)
\(\chi_{40320}(34163,\cdot)\)
\(\chi_{40320}(34667,\cdot)\)
\(\chi_{40320}(37763,\cdot)\)
\(\chi_{40320}(38267,\cdot)\)
...
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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 ]
\((8191,23941,17921,32257,28801)\) → \((-1,e\left(\frac{5}{32}\right),-1,i,e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 40320 }(4427, a) \) |
\(1\) | \(1\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) |
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sage:chi.jacobi_sum(n)
