Properties

Label 2009.312
Modulus $2009$
Conductor $287$
Order $30$
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(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([20,3]))
 
pari: [g,chi] = znchar(Mod(312,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{287}(25,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2009.bg

\(\chi_{2009}(312,\cdot)\) \(\chi_{2009}(373,\cdot)\) \(\chi_{2009}(802,\cdot)\) \(\chi_{2009}(851,\cdot)\) \(\chi_{2009}(1255,\cdot)\) \(\chi_{2009}(1439,\cdot)\) \(\chi_{2009}(1745,\cdot)\) \(\chi_{2009}(1794,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.30.2799786606968243455566601400526410348799502495997177045534681.1

Values on generators

\((493,785)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{1}{30}\right)\)
value at e.g. 2