Properties

Label 2009.2
Modulus $2009$
Conductor $2009$
Order $420$
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(420))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([260,273]))
 
pari: [g,chi] = znchar(Mod(2,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(420\)
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.cj

\(\chi_{2009}(2,\cdot)\) \(\chi_{2009}(39,\cdot)\) \(\chi_{2009}(46,\cdot)\) \(\chi_{2009}(74,\cdot)\) \(\chi_{2009}(102,\cdot)\) \(\chi_{2009}(121,\cdot)\) \(\chi_{2009}(144,\cdot)\) \(\chi_{2009}(156,\cdot)\) \(\chi_{2009}(172,\cdot)\) \(\chi_{2009}(184,\cdot)\) \(\chi_{2009}(200,\cdot)\) \(\chi_{2009}(207,\cdot)\) \(\chi_{2009}(254,\cdot)\) \(\chi_{2009}(282,\cdot)\) \(\chi_{2009}(289,\cdot)\) \(\chi_{2009}(326,\cdot)\) \(\chi_{2009}(333,\cdot)\) \(\chi_{2009}(389,\cdot)\) \(\chi_{2009}(408,\cdot)\) \(\chi_{2009}(415,\cdot)\) \(\chi_{2009}(431,\cdot)\) \(\chi_{2009}(443,\cdot)\) \(\chi_{2009}(487,\cdot)\) \(\chi_{2009}(494,\cdot)\) \(\chi_{2009}(513,\cdot)\) \(\chi_{2009}(541,\cdot)\) \(\chi_{2009}(576,\cdot)\) \(\chi_{2009}(613,\cdot)\) \(\chi_{2009}(620,\cdot)\) \(\chi_{2009}(648,\cdot)\) ...

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: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial (not computed)

Values on generators

\((493,785)\) → \((e\left(\frac{13}{21}\right),e\left(\frac{13}{20}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{209}{210}\right)\)\(e\left(\frac{31}{84}\right)\)\(e\left(\frac{104}{105}\right)\)\(e\left(\frac{53}{210}\right)\)\(e\left(\frac{51}{140}\right)\)\(e\left(\frac{69}{70}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{26}{105}\right)\)\(e\left(\frac{299}{420}\right)\)\(e\left(\frac{151}{420}\right)\)
value at e.g. 2