Properties

Label 2009.342
Modulus $2009$
Conductor $287$
Order $8$
Real no
Primitive no
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(8))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,5]))
 
pari: [g,chi] = znchar(Mod(342,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{287}(55,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2009.m

\(\chi_{2009}(342,\cdot)\) \(\chi_{2009}(489,\cdot)\) \(\chi_{2009}(1028,\cdot)\) \(\chi_{2009}(1175,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.8.467605011588281.1

Values on generators

\((493,785)\) → \((-1,e\left(\frac{5}{8}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(i\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(i\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(-i\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{3}{8}\right)\)
value at e.g. 2