Properties

Label 2009.81
Modulus $2009$
Conductor $2009$
Order $42$
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(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,21]))
 
pari: [g,chi] = znchar(Mod(81,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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.bl

\(\chi_{2009}(81,\cdot)\) \(\chi_{2009}(163,\cdot)\) \(\chi_{2009}(368,\cdot)\) \(\chi_{2009}(450,\cdot)\) \(\chi_{2009}(737,\cdot)\) \(\chi_{2009}(942,\cdot)\) \(\chi_{2009}(1024,\cdot)\) \(\chi_{2009}(1229,\cdot)\) \(\chi_{2009}(1311,\cdot)\) \(\chi_{2009}(1516,\cdot)\) \(\chi_{2009}(1803,\cdot)\) \(\chi_{2009}(1885,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((493,785)\) → \((e\left(\frac{2}{21}\right),-1)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{23}{42}\right)\)
value at e.g. 2