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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([33,0,33,11,57]))
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pari:[g,chi] = znchar(Mod(3503,7728))
\(\chi_{7728}(143,\cdot)\)
\(\chi_{7728}(383,\cdot)\)
\(\chi_{7728}(479,\cdot)\)
\(\chi_{7728}(815,\cdot)\)
\(\chi_{7728}(1055,\cdot)\)
\(\chi_{7728}(1391,\cdot)\)
\(\chi_{7728}(1487,\cdot)\)
\(\chi_{7728}(2159,\cdot)\)
\(\chi_{7728}(2399,\cdot)\)
\(\chi_{7728}(2495,\cdot)\)
\(\chi_{7728}(2735,\cdot)\)
\(\chi_{7728}(3503,\cdot)\)
\(\chi_{7728}(3743,\cdot)\)
\(\chi_{7728}(3839,\cdot)\)
\(\chi_{7728}(4847,\cdot)\)
\(\chi_{7728}(6431,\cdot)\)
\(\chi_{7728}(6767,\cdot)\)
\(\chi_{7728}(7103,\cdot)\)
\(\chi_{7728}(7439,\cdot)\)
\(\chi_{7728}(7535,\cdot)\)
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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 ]
\((4831,5797,5153,6625,6721)\) → \((-1,1,-1,e\left(\frac{1}{6}\right),e\left(\frac{19}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(3503, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) |
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sage:chi.jacobi_sum(n)
