Properties

Label 2736.331
Modulus $2736$
Conductor $2736$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2736, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,3,8,2]))
 
pari: [g,chi] = znchar(Mod(331,2736))
 

Basic properties

Modulus: \(2736\)
Conductor: \(2736\)
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 2736.dt

\(\chi_{2736}(331,\cdot)\) \(\chi_{2736}(715,\cdot)\) \(\chi_{2736}(1699,\cdot)\) \(\chi_{2736}(2083,\cdot)\)

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: 12.12.2267075282619804332597478162432.1

Values on generators

\((1711,2053,1217,1009)\) → \((-1,i,e\left(\frac{2}{3}\right),e\left(\frac{1}{6}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(i\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2736 }(331,a) \;\) at \(\;a = \) e.g. 2