Properties

Conductor 2003
Order 22
Real No
Primitive Yes
Parity Odd
Orbit Label 2003.g

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(2003)
 
sage: chi = H[180]
 
pari: [g,chi] = znchar(Mod(180,2003))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 2003
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 22
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 = Odd
Orbit label = 2003.g
Orbit index = 7

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 ]
 

\(\chi_{2003}(180,\cdot)\) \(\chi_{2003}(282,\cdot)\) \(\chi_{2003}(596,\cdot)\) \(\chi_{2003}(882,\cdot)\) \(\chi_{2003}(1117,\cdot)\) \(\chi_{2003}(1243,\cdot)\) \(\chi_{2003}(1267,\cdot)\) \(\chi_{2003}(1318,\cdot)\) \(\chi_{2003}(1480,\cdot)\) \(\chi_{2003}(1651,\cdot)\)

Values on generators

\(5\) → \(e\left(\frac{15}{22}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(-1\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{9}{22}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{11})\)