Properties

Conductor 4001
Order 500
Real No
Primitive No
Parity Even
Orbit Label 8002.t

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(8002)
sage: chi = H[49]
pari: [g,chi] = znchar(Mod(49,8002))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 4001
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 500
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 = Even
Orbit label = 8002.t
Orbit index = 20

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 ]

\(\chi_{8002}(49,\cdot)\) \(\chi_{8002}(51,\cdot)\) \(\chi_{8002}(91,\cdot)\) \(\chi_{8002}(133,\cdot)\) \(\chi_{8002}(159,\cdot)\) \(\chi_{8002}(169,\cdot)\) \(\chi_{8002}(187,\cdot)\) \(\chi_{8002}(247,\cdot)\) \(\chi_{8002}(361,\cdot)\) \(\chi_{8002}(393,\cdot)\) \(\chi_{8002}(541,\cdot)\) \(\chi_{8002}(543,\cdot)\) \(\chi_{8002}(553,\cdot)\) \(\chi_{8002}(567,\cdot)\) \(\chi_{8002}(629,\cdot)\) \(\chi_{8002}(643,\cdot)\) \(\chi_{8002}(747,\cdot)\) \(\chi_{8002}(861,\cdot)\) \(\chi_{8002}(875,\cdot)\) \(\chi_{8002}(1009,\cdot)\) \(\chi_{8002}(1027,\cdot)\) \(\chi_{8002}(1053,\cdot)\) \(\chi_{8002}(1099,\cdot)\) \(\chi_{8002}(1211,\cdot)\) \(\chi_{8002}(1297,\cdot)\) \(\chi_{8002}(1319,\cdot)\) \(\chi_{8002}(1383,\cdot)\) \(\chi_{8002}(1441,\cdot)\) \(\chi_{8002}(1457,\cdot)\) \(\chi_{8002}(1501,\cdot)\) ...

Inducing primitive character

\(\chi_{4001}(49,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{57}{500}\right)\)

Values

-113579111315171921
\(1\)\(1\)\(e\left(\frac{57}{500}\right)\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{124}{125}\right)\)\(e\left(\frac{57}{250}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{87}{125}\right)\)\(e\left(\frac{317}{500}\right)\)\(e\left(\frac{67}{500}\right)\)\(e\left(\frac{42}{125}\right)\)\(e\left(\frac{53}{500}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{500})\)