Properties

Label 2019.2018
Modulus $2019$
Conductor $2019$
Order $2$
Real yes
Primitive yes
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(2019)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,1]))
 
pari: [g,chi] = znchar(Mod(2018,2019))
 

Kronecker symbol representation

sage: kronecker_character(-2019)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{-2019}{\bullet}\right)\)

Basic properties

Modulus: \(2019\)
Conductor: \(2019\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2019.c

\(\chi_{2019}(2018,\cdot)\)

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

\((674,1351)\) → \((-1,-1)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(-1\)\(1\)\(1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)\(1\)
value at e.g. 2

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{-2019}) \)