Properties

Modulus 62
Conductor 31
Order 5
Real no
Primitive no
Minimal yes
Parity even
Orbit label 62.d

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(62)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(47,62))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 62
Conductor = 31
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 5
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 62.d
Orbit index = 4

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 ]
 

\(\chi_{62}(33,\cdot)\) \(\chi_{62}(35,\cdot)\) \(\chi_{62}(39,\cdot)\) \(\chi_{62}(47,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{1}{5}\right)\)

Values

-113579111315171921
\(1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{5}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{5})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 62 }(47,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{62}(47,\cdot)) = \sum_{r\in \Z/62\Z} \chi_{62}(47,r) e\left(\frac{r}{31}\right) = 4.5524166695+-3.2055424607i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 62 }(47,·),\chi_{ 62 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{62}(47,\cdot),\chi_{62}(1,\cdot)) = \sum_{r\in \Z/62\Z} \chi_{62}(47,r) \chi_{62}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 62 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{62}(47,·)) = \sum_{r \in \Z/62\Z} \chi_{62}(47,r) e\left(\frac{1 r + 2 r^{-1}}{62}\right) = 4.7579495475+-3.4568526924i \)