Properties

Label 3744.gh
Modulus $3744$
Conductor $156$
Order $12$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3744, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,0,6,11]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(1151,3744))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(3744\)
Conductor: \(156\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 156.v
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.0.5351362262028177408.1

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{3744}(1151,\cdot)\) \(-1\) \(1\) \(-i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{3744}(1727,\cdot)\) \(-1\) \(1\) \(-i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{3744}(2303,\cdot)\) \(-1\) \(1\) \(i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{3744}(2879,\cdot)\) \(-1\) \(1\) \(i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\)