Properties

Label 2009.19
Modulus $2009$
Conductor $287$
Order $120$
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(120))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([100,27]))
 
pari: [g,chi] = znchar(Mod(19,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{287}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2009.bx

\(\chi_{2009}(19,\cdot)\) \(\chi_{2009}(117,\cdot)\) \(\chi_{2009}(129,\cdot)\) \(\chi_{2009}(227,\cdot)\) \(\chi_{2009}(276,\cdot)\) \(\chi_{2009}(313,\cdot)\) \(\chi_{2009}(362,\cdot)\) \(\chi_{2009}(423,\cdot)\) \(\chi_{2009}(509,\cdot)\) \(\chi_{2009}(521,\cdot)\) \(\chi_{2009}(668,\cdot)\) \(\chi_{2009}(766,\cdot)\) \(\chi_{2009}(803,\cdot)\) \(\chi_{2009}(913,\cdot)\) \(\chi_{2009}(950,\cdot)\) \(\chi_{2009}(962,\cdot)\) \(\chi_{2009}(999,\cdot)\) \(\chi_{2009}(1060,\cdot)\) \(\chi_{2009}(1195,\cdot)\) \(\chi_{2009}(1256,\cdot)\) \(\chi_{2009}(1293,\cdot)\) \(\chi_{2009}(1305,\cdot)\) \(\chi_{2009}(1342,\cdot)\) \(\chi_{2009}(1452,\cdot)\) \(\chi_{2009}(1489,\cdot)\) \(\chi_{2009}(1587,\cdot)\) \(\chi_{2009}(1734,\cdot)\) \(\chi_{2009}(1746,\cdot)\) \(\chi_{2009}(1832,\cdot)\) \(\chi_{2009}(1893,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((493,785)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{9}{40}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{120}\right)\)\(e\left(\frac{29}{120}\right)\)
value at e.g. 2