Properties

Conductor 43
Order 7
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 43.e

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(43)
 
sage: chi = H[41]
 
pari: [g,chi] = znchar(Mod(41,43))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 43
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 7
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 = even
Orbit label = 43.e
Orbit index = 5

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 ]
 

\(\chi_{43}(4,\cdot)\) \(\chi_{43}(11,\cdot)\) \(\chi_{43}(16,\cdot)\) \(\chi_{43}(21,\cdot)\) \(\chi_{43}(35,\cdot)\) \(\chi_{43}(41,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{1}{7}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(1\)\(1\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{2}{7}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{7})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 43 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{43}(41,\cdot)) = \sum_{r\in \Z/43\Z} \chi_{43}(41,r) e\left(\frac{2r}{43}\right) = 6.3839752672+1.4982856164i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 43 }(41,·),\chi_{ 43 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{43}(41,\cdot),\chi_{43}(1,\cdot)) = \sum_{r\in \Z/43\Z} \chi_{43}(41,r) \chi_{43}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 43 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{43}(41,·)) = \sum_{r \in \Z/43\Z} \chi_{43}(41,r) e\left(\frac{1 r + 2 r^{-1}}{43}\right) = 5.2885052421+-2.546809896i \)