Properties

Modulus 82
Conductor 41
Order 8
Real no
Primitive no
Minimal yes
Parity odd
Orbit label 82.e

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(82)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(27,82))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 82
Conductor = 41
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 8
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 = odd
Orbit label = 82.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 ]
 

\(\chi_{82}(3,\cdot)\) \(\chi_{82}(27,\cdot)\) \(\chi_{82}(55,\cdot)\) \(\chi_{82}(79,\cdot)\)

Values on generators

\(47\) → \(e\left(\frac{1}{8}\right)\)

Values

-113579111315171921
\(-1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-i\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{8})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 82 }(27,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{82}(27,\cdot)) = \sum_{r\in \Z/82\Z} \chi_{82}(27,r) e\left(\frac{r}{41}\right) = 2.5669791857+5.8660564147i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 82 }(27,·),\chi_{ 82 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{82}(27,\cdot),\chi_{82}(1,\cdot)) = \sum_{r\in \Z/82\Z} \chi_{82}(27,r) \chi_{82}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 82 }(27,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{82}(27,·)) = \sum_{r \in \Z/82\Z} \chi_{82}(27,r) e\left(\frac{1 r + 2 r^{-1}}{82}\right) = 3.3687161623+-3.3687161623i \)