Properties

Conductor 15
Order 4
Real no
Primitive no
Minimal yes
Parity even
Orbit label 105.j

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(105)
 
sage: chi = H[8]
 
pari: [g,chi] = znchar(Mod(8,105))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 15
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 = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 105.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 ]
 

\(\chi_{105}(8,\cdot)\) \(\chi_{105}(92,\cdot)\)

Values on generators

\((71,22,31)\) → \((-1,-i,1)\)

Values

-1124811131617192223
\(1\)\(1\)\(i\)\(-1\)\(-i\)\(-1\)\(i\)\(1\)\(i\)\(-1\)\(-i\)\(-i\)
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)
 
\( \tau_{ a }( \chi_{ 105 }(8,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{105}(8,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(8,r) e\left(\frac{2r}{105}\right) = 2.0361478418+3.2945564142i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 105 }(8,·),\chi_{ 105 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{105}(8,\cdot),\chi_{105}(1,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(8,r) \chi_{105}(1,1-r) = 5 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 105 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{105}(8,·)) = \sum_{r \in \Z/105\Z} \chi_{105}(8,r) e\left(\frac{1 r + 2 r^{-1}}{105}\right) = -0.0 \)