Properties

Label 507.98
Modulus $507$
Conductor $507$
Order $156$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(507, base_ring=CyclotomicField(156))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([78,59]))
 
pari: [g,chi] = znchar(Mod(98,507))
 

Basic properties

Modulus: \(507\)
Conductor: \(507\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
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 507.x

\(\chi_{507}(2,\cdot)\) \(\chi_{507}(11,\cdot)\) \(\chi_{507}(20,\cdot)\) \(\chi_{507}(32,\cdot)\) \(\chi_{507}(41,\cdot)\) \(\chi_{507}(50,\cdot)\) \(\chi_{507}(59,\cdot)\) \(\chi_{507}(71,\cdot)\) \(\chi_{507}(98,\cdot)\) \(\chi_{507}(110,\cdot)\) \(\chi_{507}(119,\cdot)\) \(\chi_{507}(128,\cdot)\) \(\chi_{507}(137,\cdot)\) \(\chi_{507}(149,\cdot)\) \(\chi_{507}(158,\cdot)\) \(\chi_{507}(167,\cdot)\) \(\chi_{507}(176,\cdot)\) \(\chi_{507}(197,\cdot)\) \(\chi_{507}(206,\cdot)\) \(\chi_{507}(215,\cdot)\) \(\chi_{507}(227,\cdot)\) \(\chi_{507}(236,\cdot)\) \(\chi_{507}(245,\cdot)\) \(\chi_{507}(254,\cdot)\) \(\chi_{507}(266,\cdot)\) \(\chi_{507}(275,\cdot)\) \(\chi_{507}(284,\cdot)\) \(\chi_{507}(293,\cdot)\) \(\chi_{507}(305,\cdot)\) \(\chi_{507}(314,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((170,340)\) → \((-1,e\left(\frac{59}{156}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{137}{156}\right)\)\(e\left(\frac{59}{78}\right)\)\(e\left(\frac{47}{52}\right)\)\(e\left(\frac{73}{156}\right)\)\(e\left(\frac{33}{52}\right)\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{71}{156}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{28}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 507 }(98,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 507 }(98,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 507 }(98,·),\chi_{ 507 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 507 }(98,·)) \;\) at \(\; a,b = \) e.g. 1,2