Properties

Modulus 547
Conductor 547
Order 21
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 547.h

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([11]))
 
pari: [g,chi] = znchar(Mod(13,547))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 547
Conductor = 547
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 21
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.h
Orbit index = 8

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 ]
 

\(\chi_{547}(13,\cdot)\) \(\chi_{547}(14,\cdot)\) \(\chi_{547}(117,\cdot)\) \(\chi_{547}(123,\cdot)\) \(\chi_{547}(126,\cdot)\) \(\chi_{547}(169,\cdot)\) \(\chi_{547}(178,\cdot)\) \(\chi_{547}(196,\cdot)\) \(\chi_{547}(360,\cdot)\) \(\chi_{547}(427,\cdot)\) \(\chi_{547}(505,\cdot)\) \(\chi_{547}(508,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{11}{21}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{2}{3}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{21})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 547 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{547}(13,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(13,r) e\left(\frac{2r}{547}\right) = 22.1088364511+-7.6288498987i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 547 }(13,·),\chi_{ 547 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{547}(13,\cdot),\chi_{547}(1,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(13,r) \chi_{547}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 547 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{547}(13,·)) = \sum_{r \in \Z/547\Z} \chi_{547}(13,r) e\left(\frac{1 r + 2 r^{-1}}{547}\right) = 3.1820970297+-42.4621339099i \)