Properties

Label 2009.12
Modulus $2009$
Conductor $2009$
Order $840$
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(840))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([220,567]))
 
pari: [g,chi] = znchar(Mod(12,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(840\)
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.cl

\(\chi_{2009}(12,\cdot)\) \(\chi_{2009}(17,\cdot)\) \(\chi_{2009}(24,\cdot)\) \(\chi_{2009}(26,\cdot)\) \(\chi_{2009}(47,\cdot)\) \(\chi_{2009}(52,\cdot)\) \(\chi_{2009}(54,\cdot)\) \(\chi_{2009}(75,\cdot)\) \(\chi_{2009}(89,\cdot)\) \(\chi_{2009}(94,\cdot)\) \(\chi_{2009}(101,\cdot)\) \(\chi_{2009}(108,\cdot)\) \(\chi_{2009}(110,\cdot)\) \(\chi_{2009}(136,\cdot)\) \(\chi_{2009}(138,\cdot)\) \(\chi_{2009}(145,\cdot)\) \(\chi_{2009}(152,\cdot)\) \(\chi_{2009}(157,\cdot)\) \(\chi_{2009}(171,\cdot)\) \(\chi_{2009}(192,\cdot)\) \(\chi_{2009}(194,\cdot)\) \(\chi_{2009}(199,\cdot)\) \(\chi_{2009}(220,\cdot)\) \(\chi_{2009}(222,\cdot)\) \(\chi_{2009}(229,\cdot)\) \(\chi_{2009}(234,\cdot)\) \(\chi_{2009}(257,\cdot)\) \(\chi_{2009}(299,\cdot)\) \(\chi_{2009}(304,\cdot)\) \(\chi_{2009}(306,\cdot)\) ...

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: $\Q(\zeta_{840})$
Fixed field: Number field defined by a degree 840 polynomial (not computed)

Values on generators

\((493,785)\) → \((e\left(\frac{11}{42}\right),e\left(\frac{27}{40}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{151}{420}\right)\)\(e\left(\frac{65}{168}\right)\)\(e\left(\frac{151}{210}\right)\)\(e\left(\frac{187}{420}\right)\)\(e\left(\frac{209}{280}\right)\)\(e\left(\frac{11}{140}\right)\)\(e\left(\frac{65}{84}\right)\)\(e\left(\frac{169}{210}\right)\)\(e\left(\frac{421}{840}\right)\)\(e\left(\frac{89}{840}\right)\)
value at e.g. 2