Properties

Label 6003.4
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([154,84,33]))
 
pari: [g,chi] = znchar(Mod(4,6003))
 

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.dk

\(\chi_{6003}(4,\cdot)\) \(\chi_{6003}(13,\cdot)\) \(\chi_{6003}(121,\cdot)\) \(\chi_{6003}(151,\cdot)\) \(\chi_{6003}(187,\cdot)\) \(\chi_{6003}(196,\cdot)\) \(\chi_{6003}(238,\cdot)\) \(\chi_{6003}(265,\cdot)\) \(\chi_{6003}(328,\cdot)\) \(\chi_{6003}(439,\cdot)\) \(\chi_{6003}(499,\cdot)\) \(\chi_{6003}(535,\cdot)\) \(\chi_{6003}(556,\cdot)\) \(\chi_{6003}(673,\cdot)\) \(\chi_{6003}(763,\cdot)\) \(\chi_{6003}(817,\cdot)\) \(\chi_{6003}(961,\cdot)\) \(\chi_{6003}(970,\cdot)\) \(\chi_{6003}(979,\cdot)\) \(\chi_{6003}(1021,\cdot)\) \(\chi_{6003}(1024,\cdot)\) \(\chi_{6003}(1048,\cdot)\) \(\chi_{6003}(1066,\cdot)\) \(\chi_{6003}(1222,\cdot)\) \(\chi_{6003}(1231,\cdot)\) \(\chi_{6003}(1327,\cdot)\) \(\chi_{6003}(1501,\cdot)\) \(\chi_{6003}(1543,\cdot)\) \(\chi_{6003}(1570,\cdot)\) \(\chi_{6003}(1600,\cdot)\) ...

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

\((668,3133,4555)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{11}\right),e\left(\frac{1}{14}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{355}{462}\right)\)\(e\left(\frac{124}{231}\right)\)\(e\left(\frac{97}{231}\right)\)\(e\left(\frac{149}{231}\right)\)\(e\left(\frac{47}{154}\right)\)\(e\left(\frac{29}{154}\right)\)\(e\left(\frac{349}{462}\right)\)\(e\left(\frac{115}{231}\right)\)\(e\left(\frac{191}{462}\right)\)\(e\left(\frac{17}{231}\right)\)
value at e.g. 2

Related number fields

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