Properties

Conductor 112
Order 4
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 112.j

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(112)
 
sage: chi = H[27]
 
pari: [g,chi] = znchar(Mod(27,112))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 112
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 = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 112.j
Orbit index = 10

Galois orbit

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

\(\chi_{112}(27,\cdot)\) \(\chi_{112}(83,\cdot)\)

Values on generators

\((15,85,17)\) → \((-1,i,-1)\)

Values

-1135911131517192325
\(1\)\(1\)\(-i\)\(-i\)\(-1\)\(-i\)\(i\)\(-1\)\(-1\)\(-i\)\(1\)\(-1\)
value at  e.g. 2

Related number fields

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

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 112 }(27,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{112}(27,\cdot)) = \sum_{r\in \Z/112\Z} \chi_{112}(27,r) e\left(\frac{r}{56}\right) = 0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 112 }(27,·),\chi_{ 112 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{112}(27,\cdot),\chi_{112}(1,\cdot)) = \sum_{r\in \Z/112\Z} \chi_{112}(27,r) \chi_{112}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 112 }(27,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{112}(27,·)) = \sum_{r \in \Z/112\Z} \chi_{112}(27,r) e\left(\frac{1 r + 2 r^{-1}}{112}\right) = 17.6183055483+-7.2977411041i \)