Properties

Label 6003.202
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([22,36,33]))
 
pari: [g,chi] = znchar(Mod(202,6003))
 

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.ci

\(\chi_{6003}(202,\cdot)\) \(\chi_{6003}(376,\cdot)\) \(\chi_{6003}(463,\cdot)\) \(\chi_{6003}(637,\cdot)\) \(\chi_{6003}(1159,\cdot)\) \(\chi_{6003}(1246,\cdot)\) \(\chi_{6003}(1507,\cdot)\) \(\chi_{6003}(1681,\cdot)\) \(\chi_{6003}(2203,\cdot)\) \(\chi_{6003}(2290,\cdot)\) \(\chi_{6003}(2464,\cdot)\) \(\chi_{6003}(2812,\cdot)\) \(\chi_{6003}(3247,\cdot)\) \(\chi_{6003}(3508,\cdot)\) \(\chi_{6003}(4291,\cdot)\) \(\chi_{6003}(4378,\cdot)\) \(\chi_{6003}(4639,\cdot)\) \(\chi_{6003}(4813,\cdot)\) \(\chi_{6003}(5161,\cdot)\) \(\chi_{6003}(5683,\cdot)\)

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

\((668,3133,4555)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{6}{11}\right),-1)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{49}{66}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{41}{66}\right)\)\(e\left(\frac{23}{33}\right)\)
value at e.g. 2

Related number fields

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