Properties

Label 6003.47
Modulus $6003$
Conductor $261$
Order $84$
Real no
Primitive no
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([14,0,33]))
 
pari: [g,chi] = znchar(Mod(47,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(261\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{261}(47,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.cq

\(\chi_{6003}(47,\cdot)\) \(\chi_{6003}(185,\cdot)\) \(\chi_{6003}(392,\cdot)\) \(\chi_{6003}(461,\cdot)\) \(\chi_{6003}(599,\cdot)\) \(\chi_{6003}(1013,\cdot)\) \(\chi_{6003}(1220,\cdot)\) \(\chi_{6003}(1634,\cdot)\) \(\chi_{6003}(1703,\cdot)\) \(\chi_{6003}(1841,\cdot)\) \(\chi_{6003}(2048,\cdot)\) \(\chi_{6003}(2462,\cdot)\) \(\chi_{6003}(2531,\cdot)\) \(\chi_{6003}(3704,\cdot)\) \(\chi_{6003}(3773,\cdot)\) \(\chi_{6003}(4187,\cdot)\) \(\chi_{6003}(4394,\cdot)\) \(\chi_{6003}(4532,\cdot)\) \(\chi_{6003}(4601,\cdot)\) \(\chi_{6003}(5015,\cdot)\) \(\chi_{6003}(5222,\cdot)\) \(\chi_{6003}(5636,\cdot)\) \(\chi_{6003}(5774,\cdot)\) \(\chi_{6003}(5843,\cdot)\)

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

\((668,3133,4555)\) → \((e\left(\frac{1}{6}\right),1,e\left(\frac{11}{28}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{83}{84}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{79}{84}\right)\)\(e\left(\frac{5}{21}\right)\)
value at e.g. 2

Related number fields

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