Properties

Label 2009.64
Modulus $2009$
Conductor $2009$
Order $70$
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(70))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([50,63]))
 
pari: [g,chi] = znchar(Mod(64,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
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.bt

\(\chi_{2009}(64,\cdot)\) \(\chi_{2009}(113,\cdot)\) \(\chi_{2009}(127,\cdot)\) \(\chi_{2009}(351,\cdot)\) \(\chi_{2009}(400,\cdot)\) \(\chi_{2009}(414,\cdot)\) \(\chi_{2009}(435,\cdot)\) \(\chi_{2009}(701,\cdot)\) \(\chi_{2009}(722,\cdot)\) \(\chi_{2009}(925,\cdot)\) \(\chi_{2009}(974,\cdot)\) \(\chi_{2009}(988,\cdot)\) \(\chi_{2009}(1009,\cdot)\) \(\chi_{2009}(1212,\cdot)\) \(\chi_{2009}(1261,\cdot)\) \(\chi_{2009}(1296,\cdot)\) \(\chi_{2009}(1499,\cdot)\) \(\chi_{2009}(1548,\cdot)\) \(\chi_{2009}(1562,\cdot)\) \(\chi_{2009}(1583,\cdot)\) \(\chi_{2009}(1786,\cdot)\) \(\chi_{2009}(1835,\cdot)\) \(\chi_{2009}(1849,\cdot)\) \(\chi_{2009}(1870,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((493,785)\) → \((e\left(\frac{5}{7}\right),e\left(\frac{9}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{13}{70}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{19}{70}\right)\)\(e\left(\frac{11}{70}\right)\)
value at e.g. 2