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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([66,99,0,110,24]))
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pari:[g,chi] = znchar(Mod(5731,7728))
\(\chi_{7728}(187,\cdot)\)
\(\chi_{7728}(859,\cdot)\)
\(\chi_{7728}(955,\cdot)\)
\(\chi_{7728}(1291,\cdot)\)
\(\chi_{7728}(1363,\cdot)\)
\(\chi_{7728}(1531,\cdot)\)
\(\chi_{7728}(1867,\cdot)\)
\(\chi_{7728}(1963,\cdot)\)
\(\chi_{7728}(2203,\cdot)\)
\(\chi_{7728}(2371,\cdot)\)
\(\chi_{7728}(2467,\cdot)\)
\(\chi_{7728}(2539,\cdot)\)
\(\chi_{7728}(2635,\cdot)\)
\(\chi_{7728}(2707,\cdot)\)
\(\chi_{7728}(2971,\cdot)\)
\(\chi_{7728}(3307,\cdot)\)
\(\chi_{7728}(3475,\cdot)\)
\(\chi_{7728}(3643,\cdot)\)
\(\chi_{7728}(3715,\cdot)\)
\(\chi_{7728}(3811,\cdot)\)
\(\chi_{7728}(4051,\cdot)\)
\(\chi_{7728}(4723,\cdot)\)
\(\chi_{7728}(4819,\cdot)\)
\(\chi_{7728}(5155,\cdot)\)
\(\chi_{7728}(5227,\cdot)\)
\(\chi_{7728}(5395,\cdot)\)
\(\chi_{7728}(5731,\cdot)\)
\(\chi_{7728}(5827,\cdot)\)
\(\chi_{7728}(6067,\cdot)\)
\(\chi_{7728}(6235,\cdot)\)
...
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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 ]
\((4831,5797,5153,6625,6721)\) → \((-1,-i,1,e\left(\frac{5}{6}\right),e\left(\frac{2}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(5731, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{2}{11}\right)\) |
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sage:chi.jacobi_sum(n)
