Properties

Label 1152.113
Modulus $1152$
Conductor $288$
Order $24$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1152, base_ring=CyclotomicField(24))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,20]))
 
pari: [g,chi] = znchar(Mod(113,1152))
 

Basic properties

Modulus: \(1152\)
Conductor: \(288\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{288}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1152.bi

\(\chi_{1152}(113,\cdot)\) \(\chi_{1152}(209,\cdot)\) \(\chi_{1152}(401,\cdot)\) \(\chi_{1152}(497,\cdot)\) \(\chi_{1152}(689,\cdot)\) \(\chi_{1152}(785,\cdot)\) \(\chi_{1152}(977,\cdot)\) \(\chi_{1152}(1073,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{24})\)
Fixed field: 24.0.1486465269728735333725176976133731985582456832.1

Values on generators

\((127,901,641)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{13}{24}\right)\)\(1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{2}{3}\right)\)
value at e.g. 2