Properties

Modulus 86
Conductor 43
Order 42
Real no
Primitive no
Minimal yes
Parity odd
Orbit label 86.h

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([13]))
 
pari: [g,chi] = znchar(Mod(55,86))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 86
Conductor = 43
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 42
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 = 86.h
Orbit index = 8

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 ]
 

\(\chi_{86}(3,\cdot)\) \(\chi_{86}(5,\cdot)\) \(\chi_{86}(19,\cdot)\) \(\chi_{86}(29,\cdot)\) \(\chi_{86}(33,\cdot)\) \(\chi_{86}(55,\cdot)\) \(\chi_{86}(61,\cdot)\) \(\chi_{86}(63,\cdot)\) \(\chi_{86}(69,\cdot)\) \(\chi_{86}(71,\cdot)\) \(\chi_{86}(73,\cdot)\) \(\chi_{86}(77,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{13}{42}\right)\)

Values

-113579111315171921
\(-1\)\(1\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{1}{7}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 86 }(55,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{86}(55,\cdot)) = \sum_{r\in \Z/86\Z} \chi_{86}(55,r) e\left(\frac{r}{43}\right) = 1.0466301521+6.473373566i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 86 }(55,·),\chi_{ 86 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{86}(55,\cdot),\chi_{86}(1,\cdot)) = \sum_{r\in \Z/86\Z} \chi_{86}(55,r) \chi_{86}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 86 }(55,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{86}(55,·)) = \sum_{r \in \Z/86\Z} \chi_{86}(55,r) e\left(\frac{1 r + 2 r^{-1}}{86}\right) = -3.7069292769+1.7851630535i \)