Properties

Label 1344.37
Modulus $1344$
Conductor $448$
Order $48$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1344)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,27,0,16]))
 
pari: [g,chi] = znchar(Mod(37,1344))
 

Basic properties

Modulus: \(1344\)
Conductor: \(448\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{448}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1344.cz

\(\chi_{1344}(37,\cdot)\) \(\chi_{1344}(109,\cdot)\) \(\chi_{1344}(205,\cdot)\) \(\chi_{1344}(277,\cdot)\) \(\chi_{1344}(373,\cdot)\) \(\chi_{1344}(445,\cdot)\) \(\chi_{1344}(541,\cdot)\) \(\chi_{1344}(613,\cdot)\) \(\chi_{1344}(709,\cdot)\) \(\chi_{1344}(781,\cdot)\) \(\chi_{1344}(877,\cdot)\) \(\chi_{1344}(949,\cdot)\) \(\chi_{1344}(1045,\cdot)\) \(\chi_{1344}(1117,\cdot)\) \(\chi_{1344}(1213,\cdot)\) \(\chi_{1344}(1285,\cdot)\)

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

\((127,1093,449,577)\) → \((1,e\left(\frac{9}{16}\right),1,e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{35}{48}\right)\)
value at e.g. 2

Related number fields

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