![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12138, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([34,34,39]))
![Copy content]()
pari:[g,chi] = znchar(Mod(11003,12138))
\(\chi_{12138}(293,\cdot)\)
\(\chi_{12138}(965,\cdot)\)
\(\chi_{12138}(1007,\cdot)\)
\(\chi_{12138}(1679,\cdot)\)
\(\chi_{12138}(1721,\cdot)\)
\(\chi_{12138}(2393,\cdot)\)
\(\chi_{12138}(2435,\cdot)\)
\(\chi_{12138}(3107,\cdot)\)
\(\chi_{12138}(3149,\cdot)\)
\(\chi_{12138}(3821,\cdot)\)
\(\chi_{12138}(3863,\cdot)\)
\(\chi_{12138}(4535,\cdot)\)
\(\chi_{12138}(4577,\cdot)\)
\(\chi_{12138}(5249,\cdot)\)
\(\chi_{12138}(5291,\cdot)\)
\(\chi_{12138}(5963,\cdot)\)
\(\chi_{12138}(6005,\cdot)\)
\(\chi_{12138}(6677,\cdot)\)
\(\chi_{12138}(6719,\cdot)\)
\(\chi_{12138}(7391,\cdot)\)
\(\chi_{12138}(7433,\cdot)\)
\(\chi_{12138}(8105,\cdot)\)
\(\chi_{12138}(8147,\cdot)\)
\(\chi_{12138}(8819,\cdot)\)
\(\chi_{12138}(8861,\cdot)\)
\(\chi_{12138}(9533,\cdot)\)
\(\chi_{12138}(10247,\cdot)\)
\(\chi_{12138}(10289,\cdot)\)
\(\chi_{12138}(10961,\cdot)\)
\(\chi_{12138}(11003,\cdot)\)
...
![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 ]
\((8093,10405,9829)\) → \((-1,-1,e\left(\frac{39}{68}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 12138 }(11003, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{5}{68}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
