Properties

Conductor 4001
Order 1000
Real No
Primitive No
Parity Even
Orbit Label 8002.v

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[7]
pari: [g,chi] = znchar(Mod(7,8002))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 4001
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 1000
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.v
Orbit index = 22

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 ]

\(\chi_{8002}(7,\cdot)\) \(\chi_{8002}(13,\cdot)\) \(\chi_{8002}(19,\cdot)\) \(\chi_{8002}(81,\cdot)\) \(\chi_{8002}(123,\cdot)\) \(\chi_{8002}(157,\cdot)\) \(\chi_{8002}(173,\cdot)\) \(\chi_{8002}(175,\cdot)\) \(\chi_{8002}(219,\cdot)\) \(\chi_{8002}(229,\cdot)\) \(\chi_{8002}(245,\cdot)\) \(\chi_{8002}(255,\cdot)\) \(\chi_{8002}(277,\cdot)\) \(\chi_{8002}(297,\cdot)\) \(\chi_{8002}(307,\cdot)\) \(\chi_{8002}(309,\cdot)\) \(\chi_{8002}(325,\cdot)\) \(\chi_{8002}(327,\cdot)\) \(\chi_{8002}(337,\cdot)\) \(\chi_{8002}(343,\cdot)\) \(\chi_{8002}(355,\cdot)\) \(\chi_{8002}(423,\cdot)\) \(\chi_{8002}(431,\cdot)\) \(\chi_{8002}(451,\cdot)\) \(\chi_{8002}(455,\cdot)\) \(\chi_{8002}(475,\cdot)\) \(\chi_{8002}(479,\cdot)\) \(\chi_{8002}(497,\cdot)\) \(\chi_{8002}(531,\cdot)\) \(\chi_{8002}(535,\cdot)\) ...

Inducing primitive character

\(\chi_{4001}(7,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{57}{1000}\right)\)

Values

-113579111315171921
\(1\)\(1\)\(e\left(\frac{57}{1000}\right)\)\(e\left(\frac{13}{50}\right)\)\(e\left(\frac{249}{250}\right)\)\(e\left(\frac{57}{500}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{87}{250}\right)\)\(e\left(\frac{317}{1000}\right)\)\(e\left(\frac{567}{1000}\right)\)\(e\left(\frac{167}{250}\right)\)\(e\left(\frac{53}{1000}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{1000})\)