Properties

Label 2009.155
Modulus $2009$
Conductor $2009$
Order $28$
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(28))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([24,7]))
 
pari: [g,chi] = znchar(Mod(155,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
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.bd

\(\chi_{2009}(155,\cdot)\) \(\chi_{2009}(337,\cdot)\) \(\chi_{2009}(624,\cdot)\) \(\chi_{2009}(729,\cdot)\) \(\chi_{2009}(911,\cdot)\) \(\chi_{2009}(1016,\cdot)\) \(\chi_{2009}(1198,\cdot)\) \(\chi_{2009}(1303,\cdot)\) \(\chi_{2009}(1485,\cdot)\) \(\chi_{2009}(1590,\cdot)\) \(\chi_{2009}(1772,\cdot)\) \(\chi_{2009}(1877,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: Number field defined by a degree 28 polynomial

Values on generators

\((493,785)\) → \((e\left(\frac{6}{7}\right),i)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{5}{28}\right)\)
value at e.g. 2