Properties

Label 192.41
Modulus $192$
Conductor $96$
Order $8$
Real no
Primitive no
Minimal no
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([0,7,4]))
 
pari: [g,chi] = znchar(Mod(41,192))
 

Basic properties

Modulus: \(192\)
Conductor: \(96\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{96}(77,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 192.p

\(\chi_{192}(41,\cdot)\) \(\chi_{192}(89,\cdot)\) \(\chi_{192}(137,\cdot)\) \(\chi_{192}(185,\cdot)\)

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

\((127,133,65)\) → \((1,e\left(\frac{7}{8}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(1\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: 8.0.173946175488.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 192 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{192}(41,\cdot)) = \sum_{r\in \Z/192\Z} \chi_{192}(41,r) e\left(\frac{r}{96}\right) = 3.8229739416+19.2193878738i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 192 }(41,·),\chi_{ 192 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{192}(41,\cdot),\chi_{192}(1,\cdot)) = \sum_{r\in \Z/192\Z} \chi_{192}(41,r) \chi_{192}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 192 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{192}(41,·)) = \sum_{r \in \Z/192\Z} \chi_{192}(41,r) e\left(\frac{1 r + 2 r^{-1}}{192}\right) = 0.0 \)