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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,0,22,36]))
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pari:[g,chi] = znchar(Mod(4675,7728))
\(\chi_{7728}(307,\cdot)\)
\(\chi_{7728}(811,\cdot)\)
\(\chi_{7728}(979,\cdot)\)
\(\chi_{7728}(1315,\cdot)\)
\(\chi_{7728}(1651,\cdot)\)
\(\chi_{7728}(1819,\cdot)\)
\(\chi_{7728}(1987,\cdot)\)
\(\chi_{7728}(2155,\cdot)\)
\(\chi_{7728}(3163,\cdot)\)
\(\chi_{7728}(3499,\cdot)\)
\(\chi_{7728}(4171,\cdot)\)
\(\chi_{7728}(4675,\cdot)\)
\(\chi_{7728}(4843,\cdot)\)
\(\chi_{7728}(5179,\cdot)\)
\(\chi_{7728}(5515,\cdot)\)
\(\chi_{7728}(5683,\cdot)\)
\(\chi_{7728}(5851,\cdot)\)
\(\chi_{7728}(6019,\cdot)\)
\(\chi_{7728}(7027,\cdot)\)
\(\chi_{7728}(7363,\cdot)\)
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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 ]
\((4831,5797,5153,6625,6721)\) → \((-1,-i,1,-1,e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(4675, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) |
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sage:chi.jacobi_sum(n)
