Properties

Label 819.fm
Modulus $819$
Conductor $91$
Order $12$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(819, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,6,5]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(370,819))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(819\)
Conductor: \(91\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 91.bc
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.12.210845878198059013.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(8\) \(10\) \(11\) \(16\) \(17\) \(19\) \(20\)
\(\chi_{819}(370,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(i\) \(i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{819}(496,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{819}(622,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(-i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{819}(748,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(-i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{12}\right)\)