Properties

Label 507.5
Modulus $507$
Conductor $507$
Order $52$
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(52))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([26,3]))
 
pari: [g,chi] = znchar(Mod(5,507))
 

Basic properties

Modulus: \(507\)
Conductor: \(507\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
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.s

\(\chi_{507}(5,\cdot)\) \(\chi_{507}(8,\cdot)\) \(\chi_{507}(44,\cdot)\) \(\chi_{507}(47,\cdot)\) \(\chi_{507}(83,\cdot)\) \(\chi_{507}(86,\cdot)\) \(\chi_{507}(122,\cdot)\) \(\chi_{507}(125,\cdot)\) \(\chi_{507}(161,\cdot)\) \(\chi_{507}(164,\cdot)\) \(\chi_{507}(200,\cdot)\) \(\chi_{507}(203,\cdot)\) \(\chi_{507}(242,\cdot)\) \(\chi_{507}(278,\cdot)\) \(\chi_{507}(281,\cdot)\) \(\chi_{507}(317,\cdot)\) \(\chi_{507}(320,\cdot)\) \(\chi_{507}(356,\cdot)\) \(\chi_{507}(359,\cdot)\) \(\chi_{507}(395,\cdot)\) \(\chi_{507}(398,\cdot)\) \(\chi_{507}(434,\cdot)\) \(\chi_{507}(473,\cdot)\) \(\chi_{507}(476,\cdot)\)

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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\((170,340)\) → \((-1,e\left(\frac{3}{52}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{29}{52}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{1}{52}\right)\)\(e\left(\frac{9}{52}\right)\)\(e\left(\frac{35}{52}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{23}{52}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{12}{13}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 507 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{507}(5,\cdot)) = \sum_{r\in \Z/507\Z} \chi_{507}(5,r) e\left(\frac{2r}{507}\right) = -22.3767492773+2.5062106414i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 507 }(5,·),\chi_{ 507 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{507}(5,\cdot),\chi_{507}(1,\cdot)) = \sum_{r\in \Z/507\Z} \chi_{507}(5,r) \chi_{507}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 507 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{507}(5,·)) = \sum_{r \in \Z/507\Z} \chi_{507}(5,r) e\left(\frac{1 r + 2 r^{-1}}{507}\right) = 0.0 \)