Properties

Label 837.380
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([5,36]))
 
pari: [g,chi] = znchar(Mod(380,837))
 

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.cg

\(\chi_{837}(2,\cdot)\) \(\chi_{837}(47,\cdot)\) \(\chi_{837}(95,\cdot)\) \(\chi_{837}(101,\cdot)\) \(\chi_{837}(128,\cdot)\) \(\chi_{837}(140,\cdot)\) \(\chi_{837}(194,\cdot)\) \(\chi_{837}(221,\cdot)\) \(\chi_{837}(281,\cdot)\) \(\chi_{837}(326,\cdot)\) \(\chi_{837}(374,\cdot)\) \(\chi_{837}(380,\cdot)\) \(\chi_{837}(407,\cdot)\) \(\chi_{837}(419,\cdot)\) \(\chi_{837}(473,\cdot)\) \(\chi_{837}(500,\cdot)\) \(\chi_{837}(560,\cdot)\) \(\chi_{837}(605,\cdot)\) \(\chi_{837}(653,\cdot)\) \(\chi_{837}(659,\cdot)\) \(\chi_{837}(686,\cdot)\) \(\chi_{837}(698,\cdot)\) \(\chi_{837}(752,\cdot)\) \(\chi_{837}(779,\cdot)\)

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

\((218,406)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{2}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{59}{90}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{83}{90}\right)\)\(e\left(\frac{38}{45}\right)\)\(e\left(\frac{67}{90}\right)\)\(e\left(\frac{28}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 837 }(380,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 837 }(380,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 837 }(380,·),\chi_{ 837 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 837 }(380,·)) \;\) at \(\; a,b = \) e.g. 1,2