Properties

Label 112.75
Modulus $112$
Conductor $112$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(112, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,3,10]))
 
pari: [g,chi] = znchar(Mod(75,112))
 

Basic properties

Modulus: \(112\)
Conductor: \(112\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
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 112.v

\(\chi_{112}(3,\cdot)\) \(\chi_{112}(19,\cdot)\) \(\chi_{112}(59,\cdot)\) \(\chi_{112}(75,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.2426443912768913408.1

Values on generators

\((15,85,17)\) → \((-1,i,e\left(\frac{5}{6}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 112 }(75, a) \) \(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(i\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 112 }(75,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 112 }(75,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 112 }(75,·),\chi_{ 112 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 112 }(75,·)) \;\) at \(\; a,b = \) e.g. 1,2