Properties

Label 2009.3
Modulus $2009$
Conductor $2009$
Order $168$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2009, base_ring=CyclotomicField(168))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,63]))
 
pari: [g,chi] = znchar(Mod(3,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(168\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2009.cb

\(\chi_{2009}(3,\cdot)\) \(\chi_{2009}(38,\cdot)\) \(\chi_{2009}(96,\cdot)\) \(\chi_{2009}(150,\cdot)\) \(\chi_{2009}(208,\cdot)\) \(\chi_{2009}(243,\cdot)\) \(\chi_{2009}(290,\cdot)\) \(\chi_{2009}(355,\cdot)\) \(\chi_{2009}(383,\cdot)\) \(\chi_{2009}(437,\cdot)\) \(\chi_{2009}(465,\cdot)\) \(\chi_{2009}(495,\cdot)\) \(\chi_{2009}(530,\cdot)\) \(\chi_{2009}(577,\cdot)\) \(\chi_{2009}(612,\cdot)\) \(\chi_{2009}(642,\cdot)\) \(\chi_{2009}(670,\cdot)\) \(\chi_{2009}(724,\cdot)\) \(\chi_{2009}(752,\cdot)\) \(\chi_{2009}(782,\cdot)\) \(\chi_{2009}(817,\cdot)\) \(\chi_{2009}(899,\cdot)\) \(\chi_{2009}(929,\cdot)\) \(\chi_{2009}(957,\cdot)\) \(\chi_{2009}(1039,\cdot)\) \(\chi_{2009}(1069,\cdot)\) \(\chi_{2009}(1104,\cdot)\) \(\chi_{2009}(1151,\cdot)\) \(\chi_{2009}(1186,\cdot)\) \(\chi_{2009}(1216,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{168})$
Fixed field: Number field defined by a degree 168 polynomial (not computed)

Values on generators

\((493,785)\) → \((e\left(\frac{1}{42}\right),e\left(\frac{3}{8}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{31}{84}\right)\)\(e\left(\frac{109}{168}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{79}{84}\right)\)\(e\left(\frac{1}{56}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{25}{84}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{13}{168}\right)\)\(e\left(\frac{65}{168}\right)\)
value at e.g. 2