Properties

Label 2009.384
Modulus $2009$
Conductor $2009$
Order $280$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2009, base_ring=CyclotomicField(280))
 
M = H._module
 
chi = DirichletCharacter(H, M([100,259]))
 
pari: [g,chi] = znchar(Mod(384,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(280\)
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.cg

\(\chi_{2009}(6,\cdot)\) \(\chi_{2009}(13,\cdot)\) \(\chi_{2009}(34,\cdot)\) \(\chi_{2009}(69,\cdot)\) \(\chi_{2009}(76,\cdot)\) \(\chi_{2009}(104,\cdot)\) \(\chi_{2009}(111,\cdot)\) \(\chi_{2009}(153,\cdot)\) \(\chi_{2009}(181,\cdot)\) \(\chi_{2009}(188,\cdot)\) \(\chi_{2009}(216,\cdot)\) \(\chi_{2009}(258,\cdot)\) \(\chi_{2009}(265,\cdot)\) \(\chi_{2009}(272,\cdot)\) \(\chi_{2009}(300,\cdot)\) \(\chi_{2009}(321,\cdot)\) \(\chi_{2009}(335,\cdot)\) \(\chi_{2009}(356,\cdot)\) \(\chi_{2009}(363,\cdot)\) \(\chi_{2009}(384,\cdot)\) \(\chi_{2009}(398,\cdot)\) \(\chi_{2009}(468,\cdot)\) \(\chi_{2009}(475,\cdot)\) \(\chi_{2009}(503,\cdot)\) \(\chi_{2009}(545,\cdot)\) \(\chi_{2009}(552,\cdot)\) \(\chi_{2009}(559,\cdot)\) \(\chi_{2009}(580,\cdot)\) \(\chi_{2009}(608,\cdot)\) \(\chi_{2009}(622,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{280})$
Fixed field: Number field defined by a degree 280 polynomial (not computed)

Values on generators

\((493,785)\) → \((e\left(\frac{5}{14}\right),e\left(\frac{37}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 2009 }(384, a) \) \(1\)\(1\)\(e\left(\frac{47}{140}\right)\)\(e\left(\frac{13}{56}\right)\)\(e\left(\frac{47}{70}\right)\)\(e\left(\frac{99}{140}\right)\)\(e\left(\frac{159}{280}\right)\)\(e\left(\frac{1}{140}\right)\)\(e\left(\frac{13}{28}\right)\)\(e\left(\frac{3}{70}\right)\)\(e\left(\frac{17}{280}\right)\)\(e\left(\frac{253}{280}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2009 }(384,a) \;\) at \(\;a = \) e.g. 2