Properties

Label 6003.139
Modulus $6003$
Conductor $261$
Order $21$
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([7,0,15]))
 
pari: [g,chi] = znchar(Mod(139,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(261\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{261}(139,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.bh

\(\chi_{6003}(139,\cdot)\) \(\chi_{6003}(277,\cdot)\) \(\chi_{6003}(484,\cdot)\) \(\chi_{6003}(691,\cdot)\) \(\chi_{6003}(1312,\cdot)\) \(\chi_{6003}(2140,\cdot)\) \(\chi_{6003}(3244,\cdot)\) \(\chi_{6003}(4279,\cdot)\) \(\chi_{6003}(4486,\cdot)\) \(\chi_{6003}(4693,\cdot)\) \(\chi_{6003}(5245,\cdot)\) \(\chi_{6003}(5314,\cdot)\)

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

\((668,3133,4555)\) → \((e\left(\frac{1}{3}\right),1,e\left(\frac{5}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{4}{21}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 21.21.4814587615056751193058435502319478353721.1