Properties

Label 2304.49
Modulus $2304$
Conductor $576$
Order $48$
Real no
Primitive no
Minimal no
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2304)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,15,16]))
 
pari: [g,chi] = znchar(Mod(49,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(576\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{576}(373,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2304.bq

\(\chi_{2304}(49,\cdot)\) \(\chi_{2304}(241,\cdot)\) \(\chi_{2304}(337,\cdot)\) \(\chi_{2304}(529,\cdot)\) \(\chi_{2304}(625,\cdot)\) \(\chi_{2304}(817,\cdot)\) \(\chi_{2304}(913,\cdot)\) \(\chi_{2304}(1105,\cdot)\) \(\chi_{2304}(1201,\cdot)\) \(\chi_{2304}(1393,\cdot)\) \(\chi_{2304}(1489,\cdot)\) \(\chi_{2304}(1681,\cdot)\) \(\chi_{2304}(1777,\cdot)\) \(\chi_{2304}(1969,\cdot)\) \(\chi_{2304}(2065,\cdot)\) \(\chi_{2304}(2257,\cdot)\)

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

\((1279,2053,1793)\) → \((1,e\left(\frac{5}{16}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{17}{48}\right)\)\(-i\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{1}{6}\right)\)
value at e.g. 2

Related number fields

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