Properties

Modulus 144
Conductor 144
Order 12
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 144.u

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(144)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,3,2]))
 
pari: [g,chi] = znchar(Mod(11,144))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 144
Conductor = 144
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 12
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 = 144.u
Orbit index = 21

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 ]
 

\(\chi_{144}(11,\cdot)\) \(\chi_{144}(59,\cdot)\) \(\chi_{144}(83,\cdot)\) \(\chi_{144}(131,\cdot)\)

Values on generators

\((127,37,65)\) → \((-1,i,e\left(\frac{1}{6}\right))\)

Values

-11571113171923252931
\(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(-1\)\(i\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{6}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 144 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{144}(11,\cdot)) = \sum_{r\in \Z/144\Z} \chi_{144}(11,r) e\left(\frac{r}{72}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 144 }(11,·),\chi_{ 144 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{144}(11,\cdot),\chi_{144}(1,\cdot)) = \sum_{r\in \Z/144\Z} \chi_{144}(11,r) \chi_{144}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 144 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{144}(11,·)) = \sum_{r \in \Z/144\Z} \chi_{144}(11,r) e\left(\frac{1 r + 2 r^{-1}}{144}\right) = 11.8973383365+1.5663143066i \)