Properties

Label 931.141
Modulus $931$
Conductor $931$
Order $42$
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(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,7]))
 
pari: [g,chi] = znchar(Mod(141,931))
 

Basic properties

Modulus: \(931\)
Conductor: \(931\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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.br

\(\chi_{931}(8,\cdot)\) \(\chi_{931}(141,\cdot)\) \(\chi_{931}(183,\cdot)\) \(\chi_{931}(274,\cdot)\) \(\chi_{931}(316,\cdot)\) \(\chi_{931}(407,\cdot)\) \(\chi_{931}(449,\cdot)\) \(\chi_{931}(582,\cdot)\) \(\chi_{931}(673,\cdot)\) \(\chi_{931}(715,\cdot)\) \(\chi_{931}(806,\cdot)\) \(\chi_{931}(848,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.0.4012683575982151476651625353295463545613564177862358851055893007057333972430504464680793872531537508239499.1

Values on generators

\((248,344)\) → \((e\left(\frac{1}{7}\right),e\left(\frac{1}{6}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(-1\)\(1\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 931 }(141,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 931 }(141,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 931 }(141,·),\chi_{ 931 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 931 }(141,·)) \;\) at \(\; a,b = \) e.g. 1,2