Properties

Label 1110.1037
Modulus $1110$
Conductor $15$
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(1110, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,1,0]))
 
pari: [g,chi] = znchar(Mod(1037,1110))
 

Basic properties

Modulus: \(1110\)
Conductor: \(15\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{15}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1110.m

\(\chi_{1110}(593,\cdot)\) \(\chi_{1110}(1037,\cdot)\)

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: \(\Q(\zeta_{15})^+\)

Values on generators

\((371,667,631)\) → \((-1,i,1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(41\)\(43\)
\( \chi_{ 1110 }(1037, a) \) \(1\)\(1\)\(i\)\(-1\)\(-i\)\(-i\)\(-1\)\(i\)\(1\)\(1\)\(-1\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1110 }(1037,a) \;\) at \(\;a = \) e.g. 2