Properties

Label 1547.64
Modulus $1547$
Conductor $221$
Order $4$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1547, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,2,1]))
 
pari: [g,chi] = znchar(Mod(64,1547))
 

Basic properties

Modulus: \(1547\)
Conductor: \(221\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{221}(64,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1547.x

\(\chi_{1547}(64,\cdot)\) \(\chi_{1547}(701,\cdot)\)

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

Related number fields

Field of values: \(\mathbb{Q}(i)\)
Fixed field: 4.4.830297.1

Values on generators

\((885,834,547)\) → \((1,-1,i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1547 }(64, a) \) \(1\)\(1\)\(1\)\(i\)\(1\)\(-i\)\(i\)\(1\)\(-1\)\(-i\)\(i\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1547 }(64,a) \;\) at \(\;a = \) e.g. 2