Properties

Label 1224.127
Modulus $1224$
Conductor $68$
Order $8$
Real no
Primitive no
Minimal no
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([4,0,0,5]))
 
pari: [g,chi] = znchar(Mod(127,1224))
 

Basic properties

Modulus: \(1224\)
Conductor: \(68\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{68}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1224.bp

\(\chi_{1224}(127,\cdot)\) \(\chi_{1224}(559,\cdot)\) \(\chi_{1224}(631,\cdot)\) \(\chi_{1224}(1063,\cdot)\)

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

\((919,613,137,649)\) → \((-1,1,1,e\left(\frac{5}{8}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(i\)\(e\left(\frac{7}{8}\right)\)\(i\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: Number field defined by a degree %d polynomial (not computed)