Properties

Label 2304.289
Modulus $2304$
Conductor $32$
Order $8$
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([0,7,0]))
 
pari: [g,chi] = znchar(Mod(289,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(32\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{32}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2304.v

\(\chi_{2304}(289,\cdot)\) \(\chi_{2304}(865,\cdot)\) \(\chi_{2304}(1441,\cdot)\) \(\chi_{2304}(2017,\cdot)\)

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

\((1279,2053,1793)\) → \((1,e\left(\frac{7}{8}\right),1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(i\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: \(\Q(\zeta_{32})^+\)