Properties

Conductor 6003
Order 42
Real No
Primitive Yes
Parity Even
Orbit Label 6003.ca

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(6003)
 
sage: chi = H[689]
 
pari: [g,chi] = znchar(Mod(689,6003))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 6003
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 42
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = Yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 6003.ca
Orbit index = 53

Galois orbit

sage: chi.sage_character().galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{6003}(689,\cdot)\) \(\chi_{6003}(758,\cdot)\) \(\chi_{6003}(1310,\cdot)\) \(\chi_{6003}(1517,\cdot)\) \(\chi_{6003}(1724,\cdot)\) \(\chi_{6003}(2759,\cdot)\) \(\chi_{6003}(3863,\cdot)\) \(\chi_{6003}(4691,\cdot)\) \(\chi_{6003}(5312,\cdot)\) \(\chi_{6003}(5519,\cdot)\) \(\chi_{6003}(5726,\cdot)\) \(\chi_{6003}(5864,\cdot)\)

Values on generators

\((668,3133,4555)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{13}{14}\right))\)

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{1}{21}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{21})\)