Properties

Conductor 32
Order 8
Real No
Primitive No
Parity Odd
Orbit Label 4000.x

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(4000)
sage: chi = H[1251]
pari: [g,chi] = znchar(Mod(1251,4000))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 32
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 = No
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 4000.x
Orbit index = 24

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 ]

\(\chi_{4000}(251,\cdot)\) \(\chi_{4000}(1251,\cdot)\) \(\chi_{4000}(2251,\cdot)\) \(\chi_{4000}(3251,\cdot)\)

Inducing primitive character

\(\chi_{32}(3,\cdot)\)

Values on generators

\((2751,2501,1377)\) → \((-1,e\left(\frac{3}{8}\right),1)\)

Values

-1137911131719212327
\(-1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(i\)\(i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(e\left(\frac{7}{8}\right)\)
value at  e.g. 2

Related number fields

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