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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,0,55,54]))
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pari:[g,chi] = znchar(Mod(7159,7728))
\(\chi_{7728}(439,\cdot)\)
\(\chi_{7728}(535,\cdot)\)
\(\chi_{7728}(775,\cdot)\)
\(\chi_{7728}(1543,\cdot)\)
\(\chi_{7728}(1783,\cdot)\)
\(\chi_{7728}(1879,\cdot)\)
\(\chi_{7728}(2119,\cdot)\)
\(\chi_{7728}(2791,\cdot)\)
\(\chi_{7728}(2887,\cdot)\)
\(\chi_{7728}(3223,\cdot)\)
\(\chi_{7728}(3463,\cdot)\)
\(\chi_{7728}(3799,\cdot)\)
\(\chi_{7728}(3895,\cdot)\)
\(\chi_{7728}(4135,\cdot)\)
\(\chi_{7728}(4471,\cdot)\)
\(\chi_{7728}(4567,\cdot)\)
\(\chi_{7728}(4903,\cdot)\)
\(\chi_{7728}(5239,\cdot)\)
\(\chi_{7728}(5575,\cdot)\)
\(\chi_{7728}(7159,\cdot)\)
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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 ]
\((4831,5797,5153,6625,6721)\) → \((-1,-1,1,e\left(\frac{5}{6}\right),e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(7159, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) |
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sage:chi.jacobi_sum(n)
