Properties

Label 4009.4
Modulus $4009$
Conductor $4009$
Order $315$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4009, base_ring=CyclotomicField(630))
 
M = H._module
 
chi = DirichletCharacter(H, M([70,6]))
 
pari: [g,chi] = znchar(Mod(4,4009))
 

Basic properties

Modulus: \(4009\)
Conductor: \(4009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(315\)
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 4009.em

\(\chi_{4009}(4,\cdot)\) \(\chi_{4009}(6,\cdot)\) \(\chi_{4009}(9,\cdot)\) \(\chi_{4009}(16,\cdot)\) \(\chi_{4009}(24,\cdot)\) \(\chi_{4009}(36,\cdot)\) \(\chi_{4009}(44,\cdot)\) \(\chi_{4009}(66,\cdot)\) \(\chi_{4009}(81,\cdot)\) \(\chi_{4009}(99,\cdot)\) \(\chi_{4009}(139,\cdot)\) \(\chi_{4009}(176,\cdot)\) \(\chi_{4009}(215,\cdot)\) \(\chi_{4009}(256,\cdot)\) \(\chi_{4009}(264,\cdot)\) \(\chi_{4009}(289,\cdot)\) \(\chi_{4009}(310,\cdot)\) \(\chi_{4009}(347,\cdot)\) \(\chi_{4009}(365,\cdot)\) \(\chi_{4009}(415,\cdot)\) \(\chi_{4009}(484,\cdot)\) \(\chi_{4009}(491,\cdot)\) \(\chi_{4009}(500,\cdot)\) \(\chi_{4009}(541,\cdot)\) \(\chi_{4009}(548,\cdot)\) \(\chi_{4009}(576,\cdot)\) \(\chi_{4009}(594,\cdot)\) \(\chi_{4009}(670,\cdot)\) \(\chi_{4009}(689,\cdot)\) \(\chi_{4009}(726,\cdot)\) ...

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: $\Q(\zeta_{315})$
Fixed field: Number field defined by a degree 315 polynomial (not computed)

Values on generators

\((2111,1901)\) → \((e\left(\frac{1}{9}\right),e\left(\frac{1}{105}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4009 }(4, a) \) \(1\)\(1\)\(e\left(\frac{38}{315}\right)\)\(e\left(\frac{269}{315}\right)\)\(e\left(\frac{76}{315}\right)\)\(e\left(\frac{11}{315}\right)\)\(e\left(\frac{307}{315}\right)\)\(e\left(\frac{104}{105}\right)\)\(e\left(\frac{38}{105}\right)\)\(e\left(\frac{223}{315}\right)\)\(e\left(\frac{7}{45}\right)\)\(e\left(\frac{92}{105}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4009 }(4,a) \;\) at \(\;a = \) e.g. 2