Properties

Label 6003.1264
Modulus $6003$
Conductor $6003$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(6003)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,6,9]))
 
pari: [g,chi] = znchar(Mod(1264,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.x

\(\chi_{6003}(1264,\cdot)\) \(\chi_{6003}(2506,\cdot)\) \(\chi_{6003}(3265,\cdot)\) \(\chi_{6003}(4507,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((668,3133,4555)\) → \((e\left(\frac{1}{3}\right),-1,-i)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(i\)\(-i\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{3}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: Number field defined by a degree 12 polynomial