Properties

Label 2009.9
Modulus $2009$
Conductor $2009$
Order $84$
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(84))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,63]))
 
pari: [g,chi] = znchar(Mod(9,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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.bu

\(\chi_{2009}(9,\cdot)\) \(\chi_{2009}(32,\cdot)\) \(\chi_{2009}(114,\cdot)\) \(\chi_{2009}(296,\cdot)\) \(\chi_{2009}(319,\cdot)\) \(\chi_{2009}(401,\cdot)\) \(\chi_{2009}(501,\cdot)\) \(\chi_{2009}(583,\cdot)\) \(\chi_{2009}(688,\cdot)\) \(\chi_{2009}(788,\cdot)\) \(\chi_{2009}(870,\cdot)\) \(\chi_{2009}(893,\cdot)\) \(\chi_{2009}(975,\cdot)\) \(\chi_{2009}(1075,\cdot)\) \(\chi_{2009}(1180,\cdot)\) \(\chi_{2009}(1262,\cdot)\) \(\chi_{2009}(1362,\cdot)\) \(\chi_{2009}(1444,\cdot)\) \(\chi_{2009}(1467,\cdot)\) \(\chi_{2009}(1649,\cdot)\) \(\chi_{2009}(1731,\cdot)\) \(\chi_{2009}(1754,\cdot)\) \(\chi_{2009}(1836,\cdot)\) \(\chi_{2009}(1936,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

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

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{25}{84}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{13}{84}\right)\)\(e\left(\frac{65}{84}\right)\)
value at e.g. 2