Properties

Label 6003.5
Modulus $6003$
Conductor $6003$
Order $462$
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([385,21,363]))
 
pari: [g,chi] = znchar(Mod(5,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(462\)
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.dh

\(\chi_{6003}(5,\cdot)\) \(\chi_{6003}(38,\cdot)\) \(\chi_{6003}(122,\cdot)\) \(\chi_{6003}(149,\cdot)\) \(\chi_{6003}(158,\cdot)\) \(\chi_{6003}(212,\cdot)\) \(\chi_{6003}(245,\cdot)\) \(\chi_{6003}(383,\cdot)\) \(\chi_{6003}(410,\cdot)\) \(\chi_{6003}(419,\cdot)\) \(\chi_{6003}(428,\cdot)\) \(\chi_{6003}(470,\cdot)\) \(\chi_{6003}(497,\cdot)\) \(\chi_{6003}(527,\cdot)\) \(\chi_{6003}(734,\cdot)\) \(\chi_{6003}(776,\cdot)\) \(\chi_{6003}(941,\cdot)\) \(\chi_{6003}(950,\cdot)\) \(\chi_{6003}(1019,\cdot)\) \(\chi_{6003}(1049,\cdot)\) \(\chi_{6003}(1193,\cdot)\) \(\chi_{6003}(1211,\cdot)\) \(\chi_{6003}(1253,\cdot)\) \(\chi_{6003}(1256,\cdot)\) \(\chi_{6003}(1280,\cdot)\) \(\chi_{6003}(1298,\cdot)\) \(\chi_{6003}(1397,\cdot)\) \(\chi_{6003}(1454,\cdot)\) \(\chi_{6003}(1463,\cdot)\) \(\chi_{6003}(1514,\cdot)\) ...

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

\((668,3133,4555)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{22}\right),e\left(\frac{11}{14}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{164}{231}\right)\)\(e\left(\frac{97}{231}\right)\)\(e\left(\frac{115}{231}\right)\)\(e\left(\frac{289}{462}\right)\)\(e\left(\frac{10}{77}\right)\)\(e\left(\frac{16}{77}\right)\)\(e\left(\frac{409}{462}\right)\)\(e\left(\frac{103}{231}\right)\)\(e\left(\frac{155}{462}\right)\)\(e\left(\frac{194}{231}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{231})$
Fixed field: Number field defined by a degree 462 polynomial