Properties

Label 1113.1112
Modulus $1113$
Conductor $1113$
Order $2$
Real yes
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1113, base_ring=CyclotomicField(2))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,1,1]))
 
pari: [g,chi] = znchar(Mod(1112,1113))
 

Kronecker symbol representation

sage: kronecker_character(1113)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{1113}{\bullet}\right)\)

Basic properties

Modulus: \(1113\)
Conductor: \(1113\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1113.f

\(\chi_{1113}(1112,\cdot)\)

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

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{1113}) \)

Values on generators

\((743,955,1009)\) → \((-1,-1,-1)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(1\)\(1\)\(-1\)\(1\)\(-1\)\(-1\)\(-1\)\(1\)\(1\)\(1\)
value at e.g. 2