Properties

Modulus 86
Conductor 43
Order 21
Real no
Primitive no
Minimal yes
Parity even
Orbit label 86.g

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(86)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8]))
 
pari: [g,chi] = znchar(Mod(23,86))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 86
Conductor = 43
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 = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 86.g
Orbit index = 7

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 ]
 

\(\chi_{86}(9,\cdot)\) \(\chi_{86}(13,\cdot)\) \(\chi_{86}(15,\cdot)\) \(\chi_{86}(17,\cdot)\) \(\chi_{86}(23,\cdot)\) \(\chi_{86}(25,\cdot)\) \(\chi_{86}(31,\cdot)\) \(\chi_{86}(53,\cdot)\) \(\chi_{86}(57,\cdot)\) \(\chi_{86}(67,\cdot)\) \(\chi_{86}(81,\cdot)\) \(\chi_{86}(83,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{8}{21}\right)\)

Values

-113579111315171921
\(1\)\(1\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{5}{7}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 86 }(23,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{86}(23,\cdot)) = \sum_{r\in \Z/86\Z} \chi_{86}(23,r) e\left(\frac{r}{43}\right) = -6.3897565747+-1.4734350737i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 86 }(23,·),\chi_{ 86 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{86}(23,\cdot),\chi_{86}(1,\cdot)) = \sum_{r\in \Z/86\Z} \chi_{86}(23,r) \chi_{86}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 86 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{86}(23,·)) = \sum_{r \in \Z/86\Z} \chi_{86}(23,r) e\left(\frac{1 r + 2 r^{-1}}{86}\right) = 2.450234431+3.0724967945i \)