Properties

Label 4608.1153
Modulus $4608$
Conductor $16$
Order $4$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(4608, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,0]))
 
pari: [g,chi] = znchar(Mod(1153,4608))
 

Basic properties

Modulus: \(4608\)
Conductor: \(16\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{16}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4608.k

\(\chi_{4608}(1153,\cdot)\) \(\chi_{4608}(3457,\cdot)\)

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: \(\Q(\sqrt{-1}) \)
Fixed field: \(\Q(\zeta_{16})^+\)

Values on generators

\((3583,2053,4097)\) → \((1,-i,1)\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 4608 }(1153, a) \) \(1\)\(1\)\(-i\)\(-1\)\(-i\)\(i\)\(1\)\(i\)\(-1\)\(-1\)\(i\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4608 }(1153,a) \;\) at \(\;a = \) e.g. 2