Properties

Label 1224.139
Modulus $1224$
Conductor $1224$
Order $48$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,24,16,3]))
 
pari: [g,chi] = znchar(Mod(139,1224))
 

Basic properties

Modulus: \(1224\)
Conductor: \(1224\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1224.da

\(\chi_{1224}(139,\cdot)\) \(\chi_{1224}(211,\cdot)\) \(\chi_{1224}(283,\cdot)\) \(\chi_{1224}(403,\cdot)\) \(\chi_{1224}(499,\cdot)\) \(\chi_{1224}(547,\cdot)\) \(\chi_{1224}(571,\cdot)\) \(\chi_{1224}(619,\cdot)\) \(\chi_{1224}(643,\cdot)\) \(\chi_{1224}(691,\cdot)\) \(\chi_{1224}(787,\cdot)\) \(\chi_{1224}(907,\cdot)\) \(\chi_{1224}(979,\cdot)\) \(\chi_{1224}(1051,\cdot)\) \(\chi_{1224}(1195,\cdot)\) \(\chi_{1224}(1219,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((919,613,137,649)\) → \((-1,-1,e\left(\frac{1}{3}\right),e\left(\frac{1}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1224 }(139, a) \) \(1\)\(1\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{48}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{31}{48}\right)\)\(e\left(\frac{35}{48}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1224 }(139,a) \;\) at \(\;a = \) e.g. 2