Properties

Label 507.4
Modulus $507$
Conductor $169$
Order $78$
Real no
Primitive no
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(78))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(4,507))
 

Basic properties

Modulus: \(507\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{169}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 507.t

\(\chi_{507}(4,\cdot)\) \(\chi_{507}(10,\cdot)\) \(\chi_{507}(43,\cdot)\) \(\chi_{507}(49,\cdot)\) \(\chi_{507}(82,\cdot)\) \(\chi_{507}(88,\cdot)\) \(\chi_{507}(121,\cdot)\) \(\chi_{507}(127,\cdot)\) \(\chi_{507}(160,\cdot)\) \(\chi_{507}(166,\cdot)\) \(\chi_{507}(199,\cdot)\) \(\chi_{507}(205,\cdot)\) \(\chi_{507}(238,\cdot)\) \(\chi_{507}(244,\cdot)\) \(\chi_{507}(277,\cdot)\) \(\chi_{507}(283,\cdot)\) \(\chi_{507}(322,\cdot)\) \(\chi_{507}(355,\cdot)\) \(\chi_{507}(394,\cdot)\) \(\chi_{507}(400,\cdot)\) \(\chi_{507}(433,\cdot)\) \(\chi_{507}(439,\cdot)\) \(\chi_{507}(472,\cdot)\) \(\chi_{507}(478,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((170,340)\) → \((1,e\left(\frac{1}{78}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{29}{78}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{25}{78}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{34}{39}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 507 }(4,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{507}(4,\cdot)) = \sum_{r\in \Z/507\Z} \chi_{507}(4,r) e\left(\frac{2r}{507}\right) = 6.7770210058+11.0937814241i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 507 }(4,·),\chi_{ 507 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{507}(4,\cdot),\chi_{507}(1,\cdot)) = \sum_{r\in \Z/507\Z} \chi_{507}(4,r) \chi_{507}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 507 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{507}(4,·)) = \sum_{r \in \Z/507\Z} \chi_{507}(4,r) e\left(\frac{1 r + 2 r^{-1}}{507}\right) = -15.5344377824+-0.6260164419i \)