Properties

Label 86.21
Modulus $86$
Conductor $43$
Order $7$
Real no
Primitive no
Minimal yes
Parity even

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([6]))
 
pari: [g,chi] = znchar(Mod(21,86))
 

Basic properties

Modulus: \(86\)
Conductor: \(43\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(7\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{43}(21,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 86.e

\(\chi_{86}(11,\cdot)\) \(\chi_{86}(21,\cdot)\) \(\chi_{86}(35,\cdot)\) \(\chi_{86}(41,\cdot)\) \(\chi_{86}(47,\cdot)\) \(\chi_{86}(59,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\(3\) → \(e\left(\frac{6}{7}\right)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{6}{7}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{7})\)
Fixed field: 7.7.6321363049.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 86 }(21,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{86}(21,\cdot)) = \sum_{r\in \Z/86\Z} \chi_{86}(21,r) e\left(\frac{r}{43}\right) = 5.151750339+4.0570270451i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 86 }(21,·),\chi_{ 86 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{86}(21,\cdot),\chi_{86}(1,\cdot)) = \sum_{r\in \Z/86\Z} \chi_{86}(21,r) \chi_{86}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 86 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{86}(21,·)) = \sum_{r \in \Z/86\Z} \chi_{86}(21,r) e\left(\frac{1 r + 2 r^{-1}}{86}\right) = 5.4450235548+2.6221851428i \)