Properties

Label 1224.61
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([0,24,32,9]))
 
pari: [g,chi] = znchar(Mod(61,1224))
 

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.cu

\(\chi_{1224}(61,\cdot)\) \(\chi_{1224}(133,\cdot)\) \(\chi_{1224}(277,\cdot)\) \(\chi_{1224}(301,\cdot)\) \(\chi_{1224}(445,\cdot)\) \(\chi_{1224}(517,\cdot)\) \(\chi_{1224}(589,\cdot)\) \(\chi_{1224}(709,\cdot)\) \(\chi_{1224}(805,\cdot)\) \(\chi_{1224}(853,\cdot)\) \(\chi_{1224}(877,\cdot)\) \(\chi_{1224}(925,\cdot)\) \(\chi_{1224}(949,\cdot)\) \(\chi_{1224}(997,\cdot)\) \(\chi_{1224}(1093,\cdot)\) \(\chi_{1224}(1213,\cdot)\)

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

\((919,613,137,649)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{3}{16}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{35}{48}\right)\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{1}{48}\right)\)\(-1\)
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)