Properties

Label 6003.86
Modulus $6003$
Conductor $6003$
Order $66$
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([55,21,33]))
 
pari: [g,chi] = znchar(Mod(86,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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.cl

\(\chi_{6003}(86,\cdot)\) \(\chi_{6003}(434,\cdot)\) \(\chi_{6003}(608,\cdot)\) \(\chi_{6003}(695,\cdot)\) \(\chi_{6003}(1217,\cdot)\) \(\chi_{6003}(1391,\cdot)\) \(\chi_{6003}(1652,\cdot)\) \(\chi_{6003}(1739,\cdot)\) \(\chi_{6003}(2261,\cdot)\) \(\chi_{6003}(2435,\cdot)\) \(\chi_{6003}(2522,\cdot)\) \(\chi_{6003}(2696,\cdot)\) \(\chi_{6003}(3218,\cdot)\) \(\chi_{6003}(3740,\cdot)\) \(\chi_{6003}(4088,\cdot)\) \(\chi_{6003}(4262,\cdot)\) \(\chi_{6003}(4523,\cdot)\) \(\chi_{6003}(4610,\cdot)\) \(\chi_{6003}(5393,\cdot)\) \(\chi_{6003}(5654,\cdot)\)

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

\((668,3133,4555)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{7}{22}\right),-1)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{13}{66}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{29}{33}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial