Properties

Label 3744.863
Modulus $3744$
Conductor $156$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3744)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,0,2,3]))
 
pari: [g,chi] = znchar(Mod(863,3744))
 

Basic properties

Modulus: \(3744\)
Conductor: \(156\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{156}(83,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3744.bg

\(\chi_{3744}(863,\cdot)\) \(\chi_{3744}(3167,\cdot)\)

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

\((703,2341,2081,2017)\) → \((-1,1,-1,-i)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(i\)\(-i\)\(i\)\(1\)\(i\)\(-1\)\(-1\)\(-1\)\(i\)\(1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: 4.0.316368.2