from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2003, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([7]))
pari: [g,chi] = znchar(Mod(1480,2003))
Basic properties
| Modulus: | \(2003\) | |
| Conductor: | \(2003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2003.g
\(\chi_{2003}(180,\cdot)\) \(\chi_{2003}(282,\cdot)\) \(\chi_{2003}(596,\cdot)\) \(\chi_{2003}(882,\cdot)\) \(\chi_{2003}(1117,\cdot)\) \(\chi_{2003}(1243,\cdot)\) \(\chi_{2003}(1267,\cdot)\) \(\chi_{2003}(1318,\cdot)\) \(\chi_{2003}(1480,\cdot)\) \(\chi_{2003}(1651,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{11})\) |
| Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\(5\) → \(e\left(\frac{7}{22}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 2003 }(1480, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(-1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) |
sage: chi.jacobi_sum(n)