Properties

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

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(86)
 
sage: chi = H[69]
 
pari: [g,chi] = znchar(Mod(69,86))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
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.sage_character().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

\(3\) → \(e\left(\frac{17}{42}\right)\)

Values

-113579111315171921
\(-1\)\(1\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{4}{7}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{21})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 86 }(69,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{86}(69,\cdot)) = \sum_{r\in \Z/86\Z} \chi_{86}(69,r) e\left(\frac{r}{43}\right) = -1.4570625448+6.3935098921i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 86 }(69,·),\chi_{ 86 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{86}(69,\cdot),\chi_{86}(1,\cdot)) = \sum_{r\in \Z/86\Z} \chi_{86}(69,r) \chi_{86}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 86 }(69,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{86}(69,·)) = \sum_{r \in \Z/86\Z} \chi_{86}(69,r) e\left(\frac{1 r + 2 r^{-1}}{86}\right) = 0.0610418788+0.2674419455i \)