Properties

Label 2009.247
Modulus $2009$
Conductor $49$
Order $21$
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(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([26,0]))
 
pari: [g,chi] = znchar(Mod(247,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2009.z

\(\chi_{2009}(247,\cdot)\) \(\chi_{2009}(452,\cdot)\) \(\chi_{2009}(534,\cdot)\) \(\chi_{2009}(739,\cdot)\) \(\chi_{2009}(821,\cdot)\) \(\chi_{2009}(1026,\cdot)\) \(\chi_{2009}(1313,\cdot)\) \(\chi_{2009}(1395,\cdot)\) \(\chi_{2009}(1600,\cdot)\) \(\chi_{2009}(1682,\cdot)\) \(\chi_{2009}(1887,\cdot)\) \(\chi_{2009}(1969,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{49})^+\)

Values on generators

\((493,785)\) → \((e\left(\frac{13}{21}\right),1)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{17}{21}\right)\)
value at e.g. 2