![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7742, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([52,70]))
![Copy content]()
pari:[g,chi] = znchar(Mod(1047,7742))
\(\chi_{7742}(177,\cdot)\)
\(\chi_{7742}(361,\cdot)\)
\(\chi_{7742}(753,\cdot)\)
\(\chi_{7742}(863,\cdot)\)
\(\chi_{7742}(1047,\cdot)\)
\(\chi_{7742}(1059,\cdot)\)
\(\chi_{7742}(1537,\cdot)\)
\(\chi_{7742}(1843,\cdot)\)
\(\chi_{7742}(2137,\cdot)\)
\(\chi_{7742}(2419,\cdot)\)
\(\chi_{7742}(3399,\cdot)\)
\(\chi_{7742}(3803,\cdot)\)
\(\chi_{7742}(4587,\cdot)\)
\(\chi_{7742}(5175,\cdot)\)
\(\chi_{7742}(5555,\cdot)\)
\(\chi_{7742}(5653,\cdot)\)
\(\chi_{7742}(6155,\cdot)\)
\(\chi_{7742}(6351,\cdot)\)
\(\chi_{7742}(6449,\cdot)\)
\(\chi_{7742}(6633,\cdot)\)
\(\chi_{7742}(6645,\cdot)\)
\(\chi_{7742}(6731,\cdot)\)
\(\chi_{7742}(7123,\cdot)\)
\(\chi_{7742}(7319,\cdot)\)
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((2845,4901)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{35}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 7742 }(1047, a) \) |
\(1\) | \(1\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{37}{39}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
