Properties

Label 1224.11
Modulus $1224$
Conductor $1224$
Order $48$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1224)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([24,24,8,21]))
 
pari: [g,chi] = znchar(Mod(11,1224))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1224.cx

\(\chi_{1224}(11,\cdot)\) \(\chi_{1224}(131,\cdot)\) \(\chi_{1224}(227,\cdot)\) \(\chi_{1224}(275,\cdot)\) \(\chi_{1224}(299,\cdot)\) \(\chi_{1224}(347,\cdot)\) \(\chi_{1224}(371,\cdot)\) \(\chi_{1224}(419,\cdot)\) \(\chi_{1224}(515,\cdot)\) \(\chi_{1224}(635,\cdot)\) \(\chi_{1224}(707,\cdot)\) \(\chi_{1224}(779,\cdot)\) \(\chi_{1224}(923,\cdot)\) \(\chi_{1224}(947,\cdot)\) \(\chi_{1224}(1091,\cdot)\) \(\chi_{1224}(1163,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((919,613,137,649)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{16}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{17}{48}\right)\)\(e\left(\frac{37}{48}\right)\)\(-1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree %d polynomial (not computed)