Properties

Label 931.113
Modulus $931$
Conductor $931$
Order $14$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(931, base_ring=CyclotomicField(14))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,7]))
 
pari: [g,chi] = znchar(Mod(113,931))
 

Basic properties

Modulus: \(931\)
Conductor: \(931\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
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 931.ba

\(\chi_{931}(113,\cdot)\) \(\chi_{931}(379,\cdot)\) \(\chi_{931}(512,\cdot)\) \(\chi_{931}(645,\cdot)\) \(\chi_{931}(778,\cdot)\) \(\chi_{931}(911,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: 14.0.171249048453908271645616513339.1

Values on generators

\((248,344)\) → \((e\left(\frac{5}{7}\right),-1)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(-1\)\(1\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 931 }(113,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 931 }(113,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 931 }(113,·),\chi_{ 931 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 931 }(113,·)) \;\) at \(\; a,b = \) e.g. 1,2