Properties

Label 224.81
Modulus $224$
Conductor $56$
Order $6$
Real no
Primitive no
Minimal no
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(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,4]))
 
pari: [g,chi] = znchar(Mod(81,224))
 

Basic properties

Modulus: \(224\)
Conductor: \(56\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{56}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 224.t

\(\chi_{224}(81,\cdot)\) \(\chi_{224}(177,\cdot)\)

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: \(\Q(\sqrt{-3}) \)
Fixed field: 6.6.1229312.1

Values on generators

\((127,197,129)\) → \((1,-1,e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 224 }(81,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 224 }(81,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 224 }(81,·),\chi_{ 224 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 224 }(81,·)) \;\) at \(\; a,b = \) e.g. 1,2