Properties

Label 2009.27
Modulus $2009$
Conductor $2009$
Order $56$
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(56))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,7]))
 
pari: [g,chi] = znchar(Mod(27,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(56\)
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.bo

\(\chi_{2009}(27,\cdot)\) \(\chi_{2009}(55,\cdot)\) \(\chi_{2009}(167,\cdot)\) \(\chi_{2009}(202,\cdot)\) \(\chi_{2009}(314,\cdot)\) \(\chi_{2009}(454,\cdot)\) \(\chi_{2009}(601,\cdot)\) \(\chi_{2009}(629,\cdot)\) \(\chi_{2009}(741,\cdot)\) \(\chi_{2009}(776,\cdot)\) \(\chi_{2009}(888,\cdot)\) \(\chi_{2009}(916,\cdot)\) \(\chi_{2009}(1063,\cdot)\) \(\chi_{2009}(1203,\cdot)\) \(\chi_{2009}(1315,\cdot)\) \(\chi_{2009}(1350,\cdot)\) \(\chi_{2009}(1462,\cdot)\) \(\chi_{2009}(1490,\cdot)\) \(\chi_{2009}(1602,\cdot)\) \(\chi_{2009}(1637,\cdot)\) \(\chi_{2009}(1749,\cdot)\) \(\chi_{2009}(1777,\cdot)\) \(\chi_{2009}(1889,\cdot)\) \(\chi_{2009}(1924,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\((493,785)\) → \((e\left(\frac{1}{14}\right),e\left(\frac{1}{8}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{53}{56}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{3}{56}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{13}{56}\right)\)\(e\left(\frac{9}{56}\right)\)
value at e.g. 2