Properties

Label 38.5
Modulus $38$
Conductor $19$
Order $9$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(38)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8]))
 
pari: [g,chi] = znchar(Mod(5,38))
 

Basic properties

Modulus: \(38\)
Conductor: \(19\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{19}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 38.e

\(\chi_{38}(5,\cdot)\) \(\chi_{38}(9,\cdot)\) \(\chi_{38}(17,\cdot)\) \(\chi_{38}(23,\cdot)\) \(\chi_{38}(25,\cdot)\) \(\chi_{38}(35,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\(21\) → \(e\left(\frac{8}{9}\right)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\(1\)\(1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{7}{9}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: \(\Q(\zeta_{19})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 38 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{38}(5,\cdot)) = \sum_{r\in \Z/38\Z} \chi_{38}(5,r) e\left(\frac{r}{19}\right) = 2.9298208853+-3.2274060141i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 38 }(5,·),\chi_{ 38 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{38}(5,\cdot),\chi_{38}(1,\cdot)) = \sum_{r\in \Z/38\Z} \chi_{38}(5,r) \chi_{38}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 38 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{38}(5,·)) = \sum_{r \in \Z/38\Z} \chi_{38}(5,r) e\left(\frac{1 r + 2 r^{-1}}{38}\right) = 6.3778550017+-2.3213493791i \)