Properties

Label 837.8
Modulus $837$
Conductor $279$
Order $30$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(837, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([5,12]))
 
pari: [g,chi] = znchar(Mod(8,837))
 

Basic properties

Modulus: \(837\)
Conductor: \(279\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{279}(101,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 837.br

\(\chi_{837}(8,\cdot)\) \(\chi_{837}(35,\cdot)\) \(\chi_{837}(233,\cdot)\) \(\chi_{837}(287,\cdot)\) \(\chi_{837}(314,\cdot)\) \(\chi_{837}(467,\cdot)\) \(\chi_{837}(746,\cdot)\) \(\chi_{837}(791,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.0.1832893007595472877934089352635102532684868670857003486803.1

Values on generators

\((218,406)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 837 }(8,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 837 }(8,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 837 }(8,·),\chi_{ 837 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 837 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2