Properties

Label 2009.471
Modulus $2009$
Conductor $287$
Order $60$
Real no
Primitive no
Minimal no
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(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([20,51]))
 
pari: [g,chi] = znchar(Mod(471,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{287}(184,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2009.bp

\(\chi_{2009}(128,\cdot)\) \(\chi_{2009}(226,\cdot)\) \(\chi_{2009}(361,\cdot)\) \(\chi_{2009}(459,\cdot)\) \(\chi_{2009}(471,\cdot)\) \(\chi_{2009}(569,\cdot)\) \(\chi_{2009}(863,\cdot)\) \(\chi_{2009}(900,\cdot)\) \(\chi_{2009}(1194,\cdot)\) \(\chi_{2009}(1292,\cdot)\) \(\chi_{2009}(1304,\cdot)\) \(\chi_{2009}(1402,\cdot)\) \(\chi_{2009}(1537,\cdot)\) \(\chi_{2009}(1635,\cdot)\) \(\chi_{2009}(1843,\cdot)\) \(\chi_{2009}(1929,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((493,785)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{17}{20}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{37}{60}\right)\)
value at e.g. 2