Properties

Label 5445.1693
Modulus $5445$
Conductor $55$
Order $4$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5445, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,3,2]))
 
pari: [g,chi] = znchar(Mod(1693,5445))
 

Basic properties

Modulus: \(5445\)
Conductor: \(55\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{55}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5445.k

\(\chi_{5445}(1693,\cdot)\) \(\chi_{5445}(2782,\cdot)\)

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: \(\Q(\sqrt{-1}) \)
Fixed field: 4.4.15125.1

Values on generators

\((3026,4357,3511)\) → \((1,-i,-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\( \chi_{ 5445 }(1693, a) \) \(1\)\(1\)\(i\)\(-1\)\(i\)\(-i\)\(-i\)\(-1\)\(1\)\(i\)\(1\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5445 }(1693,a) \;\) at \(\;a = \) e.g. 2