Properties

Conductor 1157
Order 8
Real No
Primitive Yes
Parity Odd
Orbit Label 1157.o

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(1157)
sage: chi = H[12]
pari: [g,chi] = znchar(Mod(12,1157))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
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
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 1157.o
Orbit index = 15

Galois orbit

sage: chi.sage_character().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

\((535,92)\) → \((-1,e\left(\frac{3}{8}\right))\)

Values

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

Related number fields

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