Properties

Label 2009.57
Modulus $2009$
Conductor $2009$
Order $35$
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([60,42]))
 
pari: [g,chi] = znchar(Mod(57,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(35\)
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.bh

\(\chi_{2009}(57,\cdot)\) \(\chi_{2009}(78,\cdot)\) \(\chi_{2009}(92,\cdot)\) \(\chi_{2009}(141,\cdot)\) \(\chi_{2009}(365,\cdot)\) \(\chi_{2009}(379,\cdot)\) \(\chi_{2009}(428,\cdot)\) \(\chi_{2009}(631,\cdot)\) \(\chi_{2009}(652,\cdot)\) \(\chi_{2009}(666,\cdot)\) \(\chi_{2009}(715,\cdot)\) \(\chi_{2009}(918,\cdot)\) \(\chi_{2009}(939,\cdot)\) \(\chi_{2009}(953,\cdot)\) \(\chi_{2009}(1002,\cdot)\) \(\chi_{2009}(1205,\cdot)\) \(\chi_{2009}(1240,\cdot)\) \(\chi_{2009}(1289,\cdot)\) \(\chi_{2009}(1492,\cdot)\) \(\chi_{2009}(1513,\cdot)\) \(\chi_{2009}(1527,\cdot)\) \(\chi_{2009}(1576,\cdot)\) \(\chi_{2009}(1779,\cdot)\) \(\chi_{2009}(1800,\cdot)\)

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 35 polynomial

Values on generators

\((493,785)\) → \((e\left(\frac{6}{7}\right),e\left(\frac{3}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{31}{35}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{27}{35}\right)\)\(e\left(\frac{2}{35}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{23}{35}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{3}{35}\right)\)\(e\left(\frac{22}{35}\right)\)
value at e.g. 2