Properties

Conductor 4001
Order 250
Real No
Primitive No
Parity Even
Orbit Label 8002.r

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[65]
pari: [g,chi] = znchar(Mod(65,8002))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 4001
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 250
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.r
Orbit index = 18

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 ]

\(\chi_{8002}(65,\cdot)\) \(\chi_{8002}(95,\cdot)\) \(\chi_{8002}(107,\cdot)\) \(\chi_{8002}(211,\cdot)\) \(\chi_{8002}(279,\cdot)\) \(\chi_{8002}(557,\cdot)\) \(\chi_{8002}(737,\cdot)\) \(\chi_{8002}(967,\cdot)\) \(\chi_{8002}(1047,\cdot)\) \(\chi_{8002}(1145,\cdot)\) \(\chi_{8002}(1161,\cdot)\) \(\chi_{8002}(1275,\cdot)\) \(\chi_{8002}(1303,\cdot)\) \(\chi_{8002}(1409,\cdot)\) \(\chi_{8002}(1485,\cdot)\) \(\chi_{8002}(1669,\cdot)\) \(\chi_{8002}(1685,\cdot)\) \(\chi_{8002}(1733,\cdot)\) \(\chi_{8002}(1763,\cdot)\) \(\chi_{8002}(1789,\cdot)\) \(\chi_{8002}(1827,\cdot)\) \(\chi_{8002}(1829,\cdot)\) \(\chi_{8002}(1837,\cdot)\) \(\chi_{8002}(2003,\cdot)\) \(\chi_{8002}(2155,\cdot)\) \(\chi_{8002}(2179,\cdot)\) \(\chi_{8002}(2255,\cdot)\) \(\chi_{8002}(2275,\cdot)\) \(\chi_{8002}(2289,\cdot)\) \(\chi_{8002}(2319,\cdot)\) ...

Inducing primitive character

\(\chi_{4001}(65,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{209}{250}\right)\)

Values

-113579111315171921
\(1\)\(1\)\(e\left(\frac{209}{250}\right)\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{76}{125}\right)\)\(e\left(\frac{84}{125}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{13}{125}\right)\)\(e\left(\frac{79}{250}\right)\)\(e\left(\frac{79}{250}\right)\)\(e\left(\frac{58}{125}\right)\)\(e\left(\frac{111}{250}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{125})\)