Properties

Label 6003.41
Modulus $6003$
Conductor $6003$
Order $132$
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([110,72,33]))
 
pari: [g,chi] = znchar(Mod(41,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
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.cu

\(\chi_{6003}(41,\cdot)\) \(\chi_{6003}(104,\cdot)\) \(\chi_{6003}(128,\cdot)\) \(\chi_{6003}(302,\cdot)\) \(\chi_{6003}(650,\cdot)\) \(\chi_{6003}(887,\cdot)\) \(\chi_{6003}(974,\cdot)\) \(\chi_{6003}(1085,\cdot)\) \(\chi_{6003}(1235,\cdot)\) \(\chi_{6003}(1346,\cdot)\) \(\chi_{6003}(1409,\cdot)\) \(\chi_{6003}(1757,\cdot)\) \(\chi_{6003}(2129,\cdot)\) \(\chi_{6003}(2216,\cdot)\) \(\chi_{6003}(2279,\cdot)\) \(\chi_{6003}(2477,\cdot)\) \(\chi_{6003}(2651,\cdot)\) \(\chi_{6003}(2801,\cdot)\) \(\chi_{6003}(2975,\cdot)\) \(\chi_{6003}(2999,\cdot)\) \(\chi_{6003}(3062,\cdot)\) \(\chi_{6003}(3236,\cdot)\) \(\chi_{6003}(3521,\cdot)\) \(\chi_{6003}(3758,\cdot)\) \(\chi_{6003}(3845,\cdot)\) \(\chi_{6003}(4043,\cdot)\) \(\chi_{6003}(4106,\cdot)\) \(\chi_{6003}(4217,\cdot)\) \(\chi_{6003}(4280,\cdot)\) \(\chi_{6003}(4304,\cdot)\) ...

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

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

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{23}{132}\right)\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{131}{132}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{115}{132}\right)\)\(e\left(\frac{23}{33}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial