Properties

Label 837.679
Modulus $837$
Conductor $837$
Order $45$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(837, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,48]))
 
pari: [g,chi] = znchar(Mod(679,837))
 

Basic properties

Modulus: \(837\)
Conductor: \(837\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 837.cb

\(\chi_{837}(76,\cdot)\) \(\chi_{837}(112,\cdot)\) \(\chi_{837}(121,\cdot)\) \(\chi_{837}(133,\cdot)\) \(\chi_{837}(142,\cdot)\) \(\chi_{837}(193,\cdot)\) \(\chi_{837}(196,\cdot)\) \(\chi_{837}(268,\cdot)\) \(\chi_{837}(355,\cdot)\) \(\chi_{837}(391,\cdot)\) \(\chi_{837}(400,\cdot)\) \(\chi_{837}(412,\cdot)\) \(\chi_{837}(421,\cdot)\) \(\chi_{837}(472,\cdot)\) \(\chi_{837}(475,\cdot)\) \(\chi_{837}(547,\cdot)\) \(\chi_{837}(634,\cdot)\) \(\chi_{837}(670,\cdot)\) \(\chi_{837}(679,\cdot)\) \(\chi_{837}(691,\cdot)\) \(\chi_{837}(700,\cdot)\) \(\chi_{837}(751,\cdot)\) \(\chi_{837}(754,\cdot)\) \(\chi_{837}(826,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ 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 45 polynomial

Values on generators

\((218,406)\) → \((e\left(\frac{1}{9}\right),e\left(\frac{8}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 837 }(679, a) \) \(1\)\(1\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{34}{45}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{29}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 837 }(679,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 837 }(679,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 837 }(679,·),\chi_{ 837 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 837 }(679,·)) \;\) at \(\; a,b = \) e.g. 1,2