Properties

Conductor 43
Order 42
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 43.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(43)
 
sage: chi = H[5]
 
pari: [g,chi] = znchar(Mod(5,43))
 

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 = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 43.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 ]
 

\(\chi_{43}(3,\cdot)\) \(\chi_{43}(5,\cdot)\) \(\chi_{43}(12,\cdot)\) \(\chi_{43}(18,\cdot)\) \(\chi_{43}(19,\cdot)\) \(\chi_{43}(20,\cdot)\) \(\chi_{43}(26,\cdot)\) \(\chi_{43}(28,\cdot)\) \(\chi_{43}(29,\cdot)\) \(\chi_{43}(30,\cdot)\) \(\chi_{43}(33,\cdot)\) \(\chi_{43}(34,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{25}{42}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{6}{7}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 43 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{43}(5,\cdot)) = \sum_{r\in \Z/43\Z} \chi_{43}(5,r) e\left(\frac{2r}{43}\right) = 4.0868079695+5.1281576244i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 43 }(5,·),\chi_{ 43 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{43}(5,\cdot),\chi_{43}(1,\cdot)) = \sum_{r\in \Z/43\Z} \chi_{43}(5,r) \chi_{43}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 43 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{43}(5,·)) = \sum_{r \in \Z/43\Z} \chi_{43}(5,r) e\left(\frac{1 r + 2 r^{-1}}{43}\right) = 1.0560686096+-4.6269388967i \)