Properties

Conductor 4033
Order 36
Real No
Primitive Yes
Parity Even
Orbit Label 4033.ej

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(4033)
sage: chi = H[237]
pari: [g,chi] = znchar(Mod(237,4033))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 4033
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 36
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 = Even
Orbit label = 4033.ej
Orbit index = 114

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 ]

\(\chi_{4033}(237,\cdot)\) \(\chi_{4033}(272,\cdot)\) \(\chi_{4033}(964,\cdot)\) \(\chi_{4033}(1004,\cdot)\) \(\chi_{4033}(1201,\cdot)\) \(\chi_{4033}(1276,\cdot)\) \(\chi_{4033}(2757,\cdot)\) \(\chi_{4033}(2832,\cdot)\) \(\chi_{4033}(3029,\cdot)\) \(\chi_{4033}(3069,\cdot)\) \(\chi_{4033}(3761,\cdot)\) \(\chi_{4033}(3796,\cdot)\)

Values on generators

\((1963,2295)\) → \((e\left(\frac{13}{36}\right),e\left(\frac{25}{36}\right))\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{17}{18}\right)\)\(-1\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(1\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{17}{36}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{36})\)