Properties

Label 2009.16
Modulus $2009$
Conductor $2009$
Order $105$
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([100,126]))
 
pari: [g,chi] = znchar(Mod(16,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(105\)
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.bw

\(\chi_{2009}(16,\cdot)\) \(\chi_{2009}(37,\cdot)\) \(\chi_{2009}(51,\cdot)\) \(\chi_{2009}(100,\cdot)\) \(\chi_{2009}(221,\cdot)\) \(\chi_{2009}(242,\cdot)\) \(\chi_{2009}(256,\cdot)\) \(\chi_{2009}(303,\cdot)\) \(\chi_{2009}(305,\cdot)\) \(\chi_{2009}(338,\cdot)\) \(\chi_{2009}(387,\cdot)\) \(\chi_{2009}(529,\cdot)\) \(\chi_{2009}(543,\cdot)\) \(\chi_{2009}(590,\cdot)\) \(\chi_{2009}(592,\cdot)\) \(\chi_{2009}(611,\cdot)\) \(\chi_{2009}(625,\cdot)\) \(\chi_{2009}(674,\cdot)\) \(\chi_{2009}(795,\cdot)\) \(\chi_{2009}(816,\cdot)\) \(\chi_{2009}(830,\cdot)\) \(\chi_{2009}(877,\cdot)\) \(\chi_{2009}(879,\cdot)\) \(\chi_{2009}(898,\cdot)\) \(\chi_{2009}(1082,\cdot)\) \(\chi_{2009}(1103,\cdot)\) \(\chi_{2009}(1117,\cdot)\) \(\chi_{2009}(1164,\cdot)\) \(\chi_{2009}(1166,\cdot)\) \(\chi_{2009}(1185,\cdot)\) ...

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 105 polynomial (not computed)

Values on generators

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

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{103}{105}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{101}{105}\right)\)\(e\left(\frac{1}{105}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{104}{105}\right)\)\(e\left(\frac{89}{105}\right)\)\(e\left(\frac{46}{105}\right)\)
value at e.g. 2