Properties

Modulus 1157
Conductor 1157
Order 8
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 1157.o

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1157)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,7]))
 
pari: [g,chi] = znchar(Mod(77,1157))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 1157
Conductor = 1157
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 8
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 1157.o
Orbit index = 15

Galois orbit

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

\(\chi_{1157}(12,\cdot)\) \(\chi_{1157}(77,\cdot)\) \(\chi_{1157}(571,\cdot)\) \(\chi_{1157}(675,\cdot)\)

Values on generators

\((535,92)\) → \((-1,e\left(\frac{7}{8}\right))\)

Values

-11234567891011
\(-1\)\(1\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(1\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(-1\)\(-i\)\(i\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{8})\)