Properties

Label 1728.593
Modulus $1728$
Conductor $48$
Order $4$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1728, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,2]))
 
pari: [g,chi] = znchar(Mod(593,1728))
 

Basic properties

Modulus: \(1728\)
Conductor: \(48\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{48}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1728.j

\(\chi_{1728}(593,\cdot)\) \(\chi_{1728}(1457,\cdot)\)

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: 4.0.18432.2

Values on generators

\((703,325,1217)\) → \((1,-i,-1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(i\)\(-1\)\(i\)\(i\)\(-1\)\(i\)\(1\)\(-1\)\(-i\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1728 }(593,a) \;\) at \(\;a = \) e.g. 2