Properties

Label 91.bd
Modulus $91$
Conductor $91$
Order $12$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(91, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8,7]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(11,91))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(91\)
Conductor: \(91\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.0.10331448031704891637.2

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(8\) \(9\) \(10\) \(11\) \(12\)
\(\chi_{91}(11,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(-i\) \(1\) \(-1\) \(-i\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{91}(58,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(i\) \(1\) \(-1\) \(i\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{91}(67,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(i\) \(1\) \(-1\) \(i\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{91}(72,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(-i\) \(1\) \(-1\) \(-i\) \(e\left(\frac{1}{6}\right)\)