Properties

Label 19.18
Modulus 1919
Conductor 1919
Order 22
Real yes
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(18,19))
 

Kronecker symbol representation

sage: kronecker_character(-19)
 
pari: znchartokronecker(g,chi)
 

(19)\displaystyle\left(\frac{-19}{\bullet}\right)

Basic properties

Modulus: 1919
Conductor: 1919
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 22
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 19.b

χ19(18,)\chi_{19}(18,\cdot)

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

Related number fields

Field of values: Q\Q
Fixed field: Q(19)\Q(\sqrt{-19})

Values on generators

221-1

Values

aa 1-111223344556677889910101111
χ19(18,a) \chi_{ 19 }(18, a) 1-1111-11-1111111111-1111-111
sage: chi.jacobi_sum(n)
 
χ19(18,a)   \chi_{ 19 }(18,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ19(18,))   \tau_{ a }( \chi_{ 19 }(18,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ19(18,),χ19(n,))   J(\chi_{ 19 }(18,·),\chi_{ 19 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ19(18,))  K(a,b,\chi_{ 19 }(18,·)) \; at   a,b=\; a,b = e.g. 1,2