Properties

Label 38.3
Modulus $38$
Conductor $19$
Order $18$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(38, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([13]))
 
pari: [g,chi] = znchar(Mod(3,38))
 

Basic properties

Modulus: \(38\)
Conductor: \(19\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{19}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 38.f

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

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

Related number fields

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

Values on generators

\(21\) → \(e\left(\frac{13}{18}\right)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\(-1\)\(1\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{4}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 38 }(3,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 38 }(3,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 38 }(3,·),\chi_{ 38 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 38 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2