Properties

Conductor 19
Order 18
Real no
Primitive no
Minimal yes
Parity odd
Orbit label 38.f

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(38)
 
sage: chi = H[21]
 
pari: [g,chi] = znchar(Mod(21,38))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 19
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 18
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 = 38.f
Orbit index = 6

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 ]
 

\(\chi_{38}(3,\cdot)\) \(\chi_{38}(13,\cdot)\) \(\chi_{38}(15,\cdot)\) \(\chi_{38}(21,\cdot)\) \(\chi_{38}(29,\cdot)\) \(\chi_{38}(33,\cdot)\)

Values on generators

\(21\) → \(e\left(\frac{1}{18}\right)\)

Values

-113579111315172123
\(-1\)\(1\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{9}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{9})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 38 }(21,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{38}(21,\cdot)) = \sum_{r\in \Z/38\Z} \chi_{38}(21,r) e\left(\frac{r}{19}\right) = 0.8561529587+4.273991356i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 38 }(21,·),\chi_{ 38 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{38}(21,\cdot),\chi_{38}(1,\cdot)) = \sum_{r\in \Z/38\Z} \chi_{38}(21,r) \chi_{38}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 38 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{38}(21,·)) = \sum_{r \in \Z/38\Z} \chi_{38}(21,r) e\left(\frac{1 r + 2 r^{-1}}{38}\right) = -0.4935312484+2.7989547966i \)