Properties

Label 6003.689
Modulus $6003$
Conductor $6003$
Order $42$
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([35,21,39]))
 
pari: [g,chi] = znchar(Mod(689,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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.ca

\(\chi_{6003}(689,\cdot)\) \(\chi_{6003}(758,\cdot)\) \(\chi_{6003}(1310,\cdot)\) \(\chi_{6003}(1517,\cdot)\) \(\chi_{6003}(1724,\cdot)\) \(\chi_{6003}(2759,\cdot)\) \(\chi_{6003}(3863,\cdot)\) \(\chi_{6003}(4691,\cdot)\) \(\chi_{6003}(5312,\cdot)\) \(\chi_{6003}(5519,\cdot)\) \(\chi_{6003}(5726,\cdot)\) \(\chi_{6003}(5864,\cdot)\)

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

\((668,3133,4555)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{13}{14}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{1}{21}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial