Properties

Label 192.13
Modulus $192$
Conductor $64$
Order $16$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(192, base_ring=CyclotomicField(16))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,15,0]))
 
pari: [g,chi] = znchar(Mod(13,192))
 

Basic properties

Modulus: \(192\)
Conductor: \(64\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{64}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 192.r

\(\chi_{192}(13,\cdot)\) \(\chi_{192}(37,\cdot)\) \(\chi_{192}(61,\cdot)\) \(\chi_{192}(85,\cdot)\) \(\chi_{192}(109,\cdot)\) \(\chi_{192}(133,\cdot)\) \(\chi_{192}(157,\cdot)\) \(\chi_{192}(181,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: \(\Q(\zeta_{64})^+\)

Values on generators

\((127,133,65)\) → \((1,e\left(\frac{15}{16}\right),1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(i\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{16}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 192 }(13,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 192 }(13,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 192 }(13,·),\chi_{ 192 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 192 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2