Properties

Label 507.71
Modulus $507$
Conductor $507$
Order $156$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: 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([78,137]))
 
pari: [g,chi] = znchar(Mod(71,507))
 

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

\((170,340)\) → \((-1,e\left(\frac{137}{156}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{59}{156}\right)\)\(e\left(\frac{59}{78}\right)\)\(e\left(\frac{21}{52}\right)\)\(e\left(\frac{151}{156}\right)\)\(e\left(\frac{7}{52}\right)\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{149}{156}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{28}{39}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 507 }(71,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{507}(71,\cdot)) = \sum_{r\in \Z/507\Z} \chi_{507}(71,r) e\left(\frac{2r}{507}\right) = -12.152077246+-18.9559230481i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 507 }(71,·),\chi_{ 507 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{507}(71,\cdot),\chi_{507}(1,\cdot)) = \sum_{r\in \Z/507\Z} \chi_{507}(71,r) \chi_{507}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 507 }(71,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{507}(71,·)) = \sum_{r \in \Z/507\Z} \chi_{507}(71,r) e\left(\frac{1 r + 2 r^{-1}}{507}\right) = -0.0 \)