Properties

Label 837.41
Modulus $837$
Conductor $837$
Order $90$
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(837, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([85,42]))
 
pari: [g,chi] = znchar(Mod(41,837))
 

Basic properties

Modulus: \(837\)
Conductor: \(837\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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 837.cd

\(\chi_{837}(38,\cdot)\) \(\chi_{837}(41,\cdot)\) \(\chi_{837}(113,\cdot)\) \(\chi_{837}(200,\cdot)\) \(\chi_{837}(236,\cdot)\) \(\chi_{837}(245,\cdot)\) \(\chi_{837}(257,\cdot)\) \(\chi_{837}(266,\cdot)\) \(\chi_{837}(317,\cdot)\) \(\chi_{837}(320,\cdot)\) \(\chi_{837}(392,\cdot)\) \(\chi_{837}(479,\cdot)\) \(\chi_{837}(515,\cdot)\) \(\chi_{837}(524,\cdot)\) \(\chi_{837}(536,\cdot)\) \(\chi_{837}(545,\cdot)\) \(\chi_{837}(596,\cdot)\) \(\chi_{837}(599,\cdot)\) \(\chi_{837}(671,\cdot)\) \(\chi_{837}(758,\cdot)\) \(\chi_{837}(794,\cdot)\) \(\chi_{837}(803,\cdot)\) \(\chi_{837}(815,\cdot)\) \(\chi_{837}(824,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((218,406)\) → \((e\left(\frac{17}{18}\right),e\left(\frac{7}{15}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{13}{90}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{90}\right)\)\(e\left(\frac{31}{45}\right)\)\(e\left(\frac{29}{90}\right)\)\(e\left(\frac{26}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 837 }(41,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 837 }(41,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 837 }(41,·),\chi_{ 837 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 837 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2