Properties

Conductor 1
Order 1
Real Yes
Primitive No
Parity Even

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(10008)
 
sage: chi = H[1]
 
pari: [g,chi] = znchar(Mod(1,10008))
 

Basic properties

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

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 ]
 

\(\chi_{10008}(1,\cdot)\)

Inducing primitive character

sage: chi.primitive_character()
 
pari: znconreyconductor(g,chi,&chi0)
 
pari: chi0
 

\(\chi_{1}(1,\cdot)\)

Values on generators

sage: chi(k) for k in H.gens()
 
pari: [ chareval(g,chi,x) | x <- g.gen ] \\ value in Q/Z
 

\((4033,2225,2503,5005)\) → \((1,1,1,1)\)

First values

157111317192325293135374143474953555961656771737779838589
111111111111111111111111111111
value at  e.g. 2

Related number fields

Field of values \(\Q\)