Properties

Label 2009.8
Modulus $2009$
Conductor $2009$
Order $140$
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(140))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([120,133]))
 
pari: [g,chi] = znchar(Mod(8,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(140\)
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.bz

\(\chi_{2009}(8,\cdot)\) \(\chi_{2009}(36,\cdot)\) \(\chi_{2009}(43,\cdot)\) \(\chi_{2009}(162,\cdot)\) \(\chi_{2009}(169,\cdot)\) \(\chi_{2009}(225,\cdot)\) \(\chi_{2009}(267,\cdot)\) \(\chi_{2009}(323,\cdot)\) \(\chi_{2009}(330,\cdot)\) \(\chi_{2009}(449,\cdot)\) \(\chi_{2009}(456,\cdot)\) \(\chi_{2009}(484,\cdot)\) \(\chi_{2009}(512,\cdot)\) \(\chi_{2009}(554,\cdot)\) \(\chi_{2009}(582,\cdot)\) \(\chi_{2009}(610,\cdot)\) \(\chi_{2009}(617,\cdot)\) \(\chi_{2009}(743,\cdot)\) \(\chi_{2009}(771,\cdot)\) \(\chi_{2009}(799,\cdot)\) \(\chi_{2009}(841,\cdot)\) \(\chi_{2009}(869,\cdot)\) \(\chi_{2009}(897,\cdot)\) \(\chi_{2009}(904,\cdot)\) \(\chi_{2009}(1023,\cdot)\) \(\chi_{2009}(1058,\cdot)\) \(\chi_{2009}(1086,\cdot)\) \(\chi_{2009}(1156,\cdot)\) \(\chi_{2009}(1184,\cdot)\) \(\chi_{2009}(1191,\cdot)\) ...

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: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

\((493,785)\) → \((e\left(\frac{6}{7}\right),e\left(\frac{19}{20}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{69}{70}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{53}{70}\right)\)\(e\left(\frac{13}{140}\right)\)\(e\left(\frac{67}{70}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{19}{140}\right)\)\(e\left(\frac{11}{140}\right)\)
value at e.g. 2