Properties

Label 2009.6
Modulus $2009$
Conductor $2009$
Order $280$
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(280))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([180,7]))
 
pari: [g,chi] = znchar(Mod(6,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(280\)
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.cg

\(\chi_{2009}(6,\cdot)\) \(\chi_{2009}(13,\cdot)\) \(\chi_{2009}(34,\cdot)\) \(\chi_{2009}(69,\cdot)\) \(\chi_{2009}(76,\cdot)\) \(\chi_{2009}(104,\cdot)\) \(\chi_{2009}(111,\cdot)\) \(\chi_{2009}(153,\cdot)\) \(\chi_{2009}(181,\cdot)\) \(\chi_{2009}(188,\cdot)\) \(\chi_{2009}(216,\cdot)\) \(\chi_{2009}(258,\cdot)\) \(\chi_{2009}(265,\cdot)\) \(\chi_{2009}(272,\cdot)\) \(\chi_{2009}(300,\cdot)\) \(\chi_{2009}(321,\cdot)\) \(\chi_{2009}(335,\cdot)\) \(\chi_{2009}(356,\cdot)\) \(\chi_{2009}(363,\cdot)\) \(\chi_{2009}(384,\cdot)\) \(\chi_{2009}(398,\cdot)\) \(\chi_{2009}(468,\cdot)\) \(\chi_{2009}(475,\cdot)\) \(\chi_{2009}(503,\cdot)\) \(\chi_{2009}(545,\cdot)\) \(\chi_{2009}(552,\cdot)\) \(\chi_{2009}(559,\cdot)\) \(\chi_{2009}(580,\cdot)\) \(\chi_{2009}(608,\cdot)\) \(\chi_{2009}(622,\cdot)\) ...

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: $\Q(\zeta_{280})$
Fixed field: Number field defined by a degree 280 polynomial (not computed)

Values on generators

\((493,785)\) → \((e\left(\frac{9}{14}\right),e\left(\frac{1}{40}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{51}{140}\right)\)\(e\left(\frac{1}{56}\right)\)\(e\left(\frac{51}{70}\right)\)\(e\left(\frac{27}{140}\right)\)\(e\left(\frac{107}{280}\right)\)\(e\left(\frac{13}{140}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{39}{70}\right)\)\(e\left(\frac{221}{280}\right)\)\(e\left(\frac{209}{280}\right)\)
value at e.g. 2