Properties

Modulus 67
Conductor 67
Order 11
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 67.e

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(67)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6]))
 
pari: [g,chi] = znchar(Mod(59,67))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 67
Conductor = 67
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 11
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 = 67.e
Orbit index = 5

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 ]
 

\(\chi_{67}(9,\cdot)\) \(\chi_{67}(14,\cdot)\) \(\chi_{67}(15,\cdot)\) \(\chi_{67}(22,\cdot)\) \(\chi_{67}(24,\cdot)\) \(\chi_{67}(25,\cdot)\) \(\chi_{67}(40,\cdot)\) \(\chi_{67}(59,\cdot)\) \(\chi_{67}(62,\cdot)\) \(\chi_{67}(64,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{6}{11}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{2}{11}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{11})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 67 }(59,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{67}(59,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(59,r) e\left(\frac{2r}{67}\right) = 0.1892070065+-8.1831656899i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 67 }(59,·),\chi_{ 67 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{67}(59,\cdot),\chi_{67}(1,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(59,r) \chi_{67}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 67 }(59,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{67}(59,·)) = \sum_{r \in \Z/67\Z} \chi_{67}(59,r) e\left(\frac{1 r + 2 r^{-1}}{67}\right) = -1.144641605+7.9611572314i \)