Properties

Label 224.3
Modulus $224$
Conductor $224$
Order $24$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(224, base_ring=CyclotomicField(24))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([12,9,4]))
 
pari: [g,chi] = znchar(Mod(3,224))
 

Basic properties

Modulus: \(224\)
Conductor: \(224\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
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 224.be

\(\chi_{224}(3,\cdot)\) \(\chi_{224}(19,\cdot)\) \(\chi_{224}(59,\cdot)\) \(\chi_{224}(75,\cdot)\) \(\chi_{224}(115,\cdot)\) \(\chi_{224}(131,\cdot)\) \(\chi_{224}(171,\cdot)\) \(\chi_{224}(187,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: 24.24.790224330201082600125157415256880139617697792.1

Values on generators

\((127,197,129)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{1}{6}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{1}{8}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 224 }(3,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 224 }(3,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 224 }(3,·),\chi_{ 224 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 224 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2