Properties

Label 931.655
Modulus $931$
Conductor $133$
Order $9$
Real no
Primitive no
Minimal no
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(931, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([12,8]))
 
pari: [g,chi] = znchar(Mod(655,931))
 

Basic properties

Modulus: \(931\)
Conductor: \(133\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{133}(123,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 931.x

\(\chi_{931}(226,\cdot)\) \(\chi_{931}(557,\cdot)\) \(\chi_{931}(655,\cdot)\) \(\chi_{931}(765,\cdot)\) \(\chi_{931}(802,\cdot)\) \(\chi_{931}(814,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 9.9.1998099208210609.1

Values on generators

\((248,344)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{4}{9}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 931 }(655, a) \) \(1\)\(1\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 931 }(655,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 931 }(655,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 931 }(655,·),\chi_{ 931 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 931 }(655,·)) \;\) at \(\; a,b = \) e.g. 1,2