Properties

Label 192.19
Modulus $192$
Conductor $64$
Order $16$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(192)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8,7,0]))
 
pari: [g,chi] = znchar(Mod(19,192))
 

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 \(\chi_{64}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 192.t

\(\chi_{192}(19,\cdot)\) \(\chi_{192}(43,\cdot)\) \(\chi_{192}(67,\cdot)\) \(\chi_{192}(91,\cdot)\) \(\chi_{192}(115,\cdot)\) \(\chi_{192}(139,\cdot)\) \(\chi_{192}(163,\cdot)\) \(\chi_{192}(187,\cdot)\)

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

\((127,133,65)\) → \((-1,e\left(\frac{7}{16}\right),1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(i\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{16})\)
Fixed field: 16.0.604462909807314587353088.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 192 }(19,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{192}(19,\cdot)) = \sum_{r\in \Z/192\Z} \chi_{192}(19,r) e\left(\frac{r}{96}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 192 }(19,·),\chi_{ 192 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{192}(19,\cdot),\chi_{192}(1,\cdot)) = \sum_{r\in \Z/192\Z} \chi_{192}(19,r) \chi_{192}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 192 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{192}(19,·)) = \sum_{r \in \Z/192\Z} \chi_{192}(19,r) e\left(\frac{1 r + 2 r^{-1}}{192}\right) = -15.9229556268+1.5682742453i \)