Properties

Label 1224.1171
Modulus $1224$
Conductor $136$
Order $8$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,4,0,3]))
 
pari: [g,chi] = znchar(Mod(1171,1224))
 

Basic properties

Modulus: \(1224\)
Conductor: \(136\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{136}(83,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1224.bv

\(\chi_{1224}(19,\cdot)\) \(\chi_{1224}(451,\cdot)\) \(\chi_{1224}(739,\cdot)\) \(\chi_{1224}(1171,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.0.1680747204608.1

Values on generators

\((919,613,137,649)\) → \((-1,-1,1,e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1224 }(1171, a) \) \(-1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(1\)\(i\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1224 }(1171,a) \;\) at \(\;a = \) e.g. 2