Properties

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

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[344]
 
pari: [g,chi] = znchar(Mod(344,6003))
 

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.bw
Orbit index = 49

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 ]
 

\(\chi_{6003}(344,\cdot)\) \(\chi_{6003}(1379,\cdot)\) \(\chi_{6003}(1586,\cdot)\) \(\chi_{6003}(1793,\cdot)\) \(\chi_{6003}(2345,\cdot)\) \(\chi_{6003}(2414,\cdot)\) \(\chi_{6003}(3242,\cdot)\) \(\chi_{6003}(3380,\cdot)\) \(\chi_{6003}(3587,\cdot)\) \(\chi_{6003}(3794,\cdot)\) \(\chi_{6003}(4415,\cdot)\) \(\chi_{6003}(5243,\cdot)\)

Values on generators

\((668,3133,4555)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{4}{7}\right))\)

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{20}{21}\right)\)
value at  e.g. 2

Related number fields

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