Properties

Modulus 160
Conductor 80
Order 4
Real no
Primitive no
Minimal no
Parity even
Orbit label 160.q

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(160)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,2]))
 
pari: [g,chi] = znchar(Mod(9,160))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 160
Conductor = 80
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 4
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = no
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 160.q
Orbit index = 17

Galois orbit

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

\(\chi_{160}(9,\cdot)\) \(\chi_{160}(89,\cdot)\)

Values on generators

\((31,101,97)\) → \((1,-i,-1)\)

Values

-1137911131719212327
\(1\)\(1\)\(-i\)\(1\)\(-1\)\(-i\)\(-i\)\(-1\)\(i\)\(-i\)\(1\)\(i\)
value at  e.g. 2

Related number fields

Field of values \(\Q(i)\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 160 }(9,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{160}(9,\cdot)) = \sum_{r\in \Z/160\Z} \chi_{160}(9,r) e\left(\frac{r}{80}\right) = 6.8456493491+-16.5268595017i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 160 }(9,·),\chi_{ 160 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{160}(9,\cdot),\chi_{160}(1,\cdot)) = \sum_{r\in \Z/160\Z} \chi_{160}(9,r) \chi_{160}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 160 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{160}(9,·)) = \sum_{r \in \Z/160\Z} \chi_{160}(9,r) e\left(\frac{1 r + 2 r^{-1}}{160}\right) = -0.0 \)