Properties

Label 1224.905
Modulus $1224$
Conductor $153$
Order $12$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,10,9]))
 
pari: [g,chi] = znchar(Mod(905,1224))
 

Basic properties

Modulus: \(1224\)
Conductor: \(153\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{153}(140,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1224.bw

\(\chi_{1224}(353,\cdot)\) \(\chi_{1224}(497,\cdot)\) \(\chi_{1224}(761,\cdot)\) \(\chi_{1224}(905,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.0.45943373101939347033.1

Values on generators

\((919,613,137,649)\) → \((1,1,e\left(\frac{5}{6}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1224 }(905, a) \) \(-1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1224 }(905,a) \;\) at \(\;a = \) e.g. 2