Properties

Label 6003.760
Modulus $6003$
Conductor $261$
Order $42$
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([14,0,9]))
 
pari: [g,chi] = znchar(Mod(760,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(261\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{261}(238,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.bx

\(\chi_{6003}(760,\cdot)\) \(\chi_{6003}(1588,\cdot)\) \(\chi_{6003}(2209,\cdot)\) \(\chi_{6003}(2416,\cdot)\) \(\chi_{6003}(2623,\cdot)\) \(\chi_{6003}(2761,\cdot)\) \(\chi_{6003}(3589,\cdot)\) \(\chi_{6003}(3658,\cdot)\) \(\chi_{6003}(4210,\cdot)\) \(\chi_{6003}(4417,\cdot)\) \(\chi_{6003}(4624,\cdot)\) \(\chi_{6003}(5659,\cdot)\)

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

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

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{4}{21}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.42.565343212441678035532894502003808167878401992443661947648452445739810658542578516149.1