Properties

Label 6003.160
Modulus $6003$
Conductor $6003$
Order $84$
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([56,42,81]))
 
pari: [g,chi] = znchar(Mod(160,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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.cp

\(\chi_{6003}(160,\cdot)\) \(\chi_{6003}(229,\cdot)\) \(\chi_{6003}(367,\cdot)\) \(\chi_{6003}(781,\cdot)\) \(\chi_{6003}(988,\cdot)\) \(\chi_{6003}(1402,\cdot)\) \(\chi_{6003}(1471,\cdot)\) \(\chi_{6003}(1609,\cdot)\) \(\chi_{6003}(1816,\cdot)\) \(\chi_{6003}(2230,\cdot)\) \(\chi_{6003}(2299,\cdot)\) \(\chi_{6003}(3472,\cdot)\) \(\chi_{6003}(3541,\cdot)\) \(\chi_{6003}(3955,\cdot)\) \(\chi_{6003}(4162,\cdot)\) \(\chi_{6003}(4300,\cdot)\) \(\chi_{6003}(4369,\cdot)\) \(\chi_{6003}(4783,\cdot)\) \(\chi_{6003}(4990,\cdot)\) \(\chi_{6003}(5404,\cdot)\) \(\chi_{6003}(5542,\cdot)\) \(\chi_{6003}(5611,\cdot)\) \(\chi_{6003}(5818,\cdot)\) \(\chi_{6003}(5956,\cdot)\)

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

\((668,3133,4555)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{27}{28}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{53}{84}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{23}{84}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{31}{84}\right)\)\(e\left(\frac{11}{21}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial