Properties

Label 1224.23
Modulus $1224$
Conductor $612$
Order $48$
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([24,0,40,45]))
 
pari: [g,chi] = znchar(Mod(23,1224))
 

Basic properties

Modulus: \(1224\)
Conductor: \(612\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{612}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1224.cy

\(\chi_{1224}(23,\cdot)\) \(\chi_{1224}(95,\cdot)\) \(\chi_{1224}(167,\cdot)\) \(\chi_{1224}(311,\cdot)\) \(\chi_{1224}(335,\cdot)\) \(\chi_{1224}(479,\cdot)\) \(\chi_{1224}(551,\cdot)\) \(\chi_{1224}(623,\cdot)\) \(\chi_{1224}(743,\cdot)\) \(\chi_{1224}(839,\cdot)\) \(\chi_{1224}(887,\cdot)\) \(\chi_{1224}(911,\cdot)\) \(\chi_{1224}(959,\cdot)\) \(\chi_{1224}(983,\cdot)\) \(\chi_{1224}(1031,\cdot)\) \(\chi_{1224}(1127,\cdot)\)

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

\((919,613,137,649)\) → \((-1,1,e\left(\frac{5}{6}\right),e\left(\frac{15}{16}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{41}{48}\right)\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{35}{48}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{1}{48}\right)\)\(e\left(\frac{29}{48}\right)\)\(1\)
value at e.g. 2

Related number fields

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