Properties

Label 1344.85
Modulus $1344$
Conductor $64$
Order $16$
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([0,13,0,0]))
 
pari: [g,chi] = znchar(Mod(85,1344))
 

Basic properties

Modulus: \(1344\)
Conductor: \(64\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{64}(21,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1344.cg

\(\chi_{1344}(85,\cdot)\) \(\chi_{1344}(253,\cdot)\) \(\chi_{1344}(421,\cdot)\) \(\chi_{1344}(589,\cdot)\) \(\chi_{1344}(757,\cdot)\) \(\chi_{1344}(925,\cdot)\) \(\chi_{1344}(1093,\cdot)\) \(\chi_{1344}(1261,\cdot)\)

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

\((127,1093,449,577)\) → \((1,e\left(\frac{13}{16}\right),1,1)\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(-i\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{15}{16}\right)\)\(-1\)\(e\left(\frac{5}{16}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{16})\)
Fixed field: \(\Q(\zeta_{64})^+\)