Properties

Label 2009.4
Modulus $2009$
Conductor $2009$
Order $210$
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(210))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([50,63]))
 
pari: [g,chi] = znchar(Mod(4,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
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.cd

\(\chi_{2009}(4,\cdot)\) \(\chi_{2009}(23,\cdot)\) \(\chi_{2009}(25,\cdot)\) \(\chi_{2009}(72,\cdot)\) \(\chi_{2009}(86,\cdot)\) \(\chi_{2009}(107,\cdot)\) \(\chi_{2009}(228,\cdot)\) \(\chi_{2009}(277,\cdot)\) \(\chi_{2009}(291,\cdot)\) \(\chi_{2009}(310,\cdot)\) \(\chi_{2009}(359,\cdot)\) \(\chi_{2009}(394,\cdot)\) \(\chi_{2009}(515,\cdot)\) \(\chi_{2009}(564,\cdot)\) \(\chi_{2009}(578,\cdot)\) \(\chi_{2009}(597,\cdot)\) \(\chi_{2009}(599,\cdot)\) \(\chi_{2009}(646,\cdot)\) \(\chi_{2009}(660,\cdot)\) \(\chi_{2009}(681,\cdot)\) \(\chi_{2009}(865,\cdot)\) \(\chi_{2009}(884,\cdot)\) \(\chi_{2009}(886,\cdot)\) \(\chi_{2009}(933,\cdot)\) \(\chi_{2009}(947,\cdot)\) \(\chi_{2009}(968,\cdot)\) \(\chi_{2009}(1089,\cdot)\) \(\chi_{2009}(1138,\cdot)\) \(\chi_{2009}(1152,\cdot)\) \(\chi_{2009}(1171,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((493,785)\) → \((e\left(\frac{5}{21}\right),e\left(\frac{3}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{104}{105}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{103}{105}\right)\)\(e\left(\frac{53}{105}\right)\)\(e\left(\frac{51}{70}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{52}{105}\right)\)\(e\left(\frac{89}{210}\right)\)\(e\left(\frac{151}{210}\right)\)
value at e.g. 2