Properties

Label 720.229
Modulus $720$
Conductor $720$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(720, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,4,6]))
 
pari: [g,chi] = znchar(Mod(229,720))
 

Basic properties

Modulus: \(720\)
Conductor: \(720\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 720.ce

\(\chi_{720}(229,\cdot)\) \(\chi_{720}(349,\cdot)\) \(\chi_{720}(589,\cdot)\) \(\chi_{720}(709,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.12.5777633090469888000000.1

Values on generators

\((271,181,641,577)\) → \((1,i,e\left(\frac{1}{3}\right),-1)\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(-1\)\(-i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(-i\)\(e\left(\frac{1}{6}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 720 }(229,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{720}(229,\cdot)) = \sum_{r\in \Z/720\Z} \chi_{720}(229,r) e\left(\frac{r}{360}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 720 }(229,·),\chi_{ 720 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{720}(229,\cdot),\chi_{720}(1,\cdot)) = \sum_{r\in \Z/720\Z} \chi_{720}(229,r) \chi_{720}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 720 }(229,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{720}(229,·)) = \sum_{r \in \Z/720\Z} \chi_{720}(229,r) e\left(\frac{1 r + 2 r^{-1}}{720}\right) = 0.0 \)