Properties

Modulus 547
Conductor 547
Order 13
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 547.f

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(547)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([12]))
 
pari: [g,chi] = znchar(Mod(509,547))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 547
Conductor = 547
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 13
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 547.f
Orbit index = 6

Galois orbit

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{547}(46,\cdot)\) \(\chi_{547}(237,\cdot)\) \(\chi_{547}(261,\cdot)\) \(\chi_{547}(293,\cdot)\) \(\chi_{547}(350,\cdot)\) \(\chi_{547}(353,\cdot)\) \(\chi_{547}(375,\cdot)\) \(\chi_{547}(440,\cdot)\) \(\chi_{547}(475,\cdot)\) \(\chi_{547}(509,\cdot)\) \(\chi_{547}(517,\cdot)\) \(\chi_{547}(519,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{12}{13}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{12}{13}\right)\)\(1\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{10}{13}\right)\)\(1\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{8}{13}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{13})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 547 }(509,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{547}(509,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(509,r) e\left(\frac{2r}{547}\right) = 23.06782726+3.8568569458i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 547 }(509,·),\chi_{ 547 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{547}(509,\cdot),\chi_{547}(1,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(509,r) \chi_{547}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 547 }(509,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{547}(509,·)) = \sum_{r \in \Z/547\Z} \chi_{547}(509,r) e\left(\frac{1 r + 2 r^{-1}}{547}\right) = 36.0876671051+-8.8948110699i \)