Properties

Label 4009.60
Modulus $4009$
Conductor $4009$
Order $630$
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([455,531]))
 
pari: [g,chi] = znchar(Mod(60,4009))
 

Basic properties

Modulus: \(4009\)
Conductor: \(4009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(630\)
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.eu

\(\chi_{4009}(60,\cdot)\) \(\chi_{4009}(86,\cdot)\) \(\chi_{4009}(89,\cdot)\) \(\chi_{4009}(90,\cdot)\) \(\chi_{4009}(97,\cdot)\) \(\chi_{4009}(98,\cdot)\) \(\chi_{4009}(124,\cdot)\) \(\chi_{4009}(129,\cdot)\) \(\chi_{4009}(135,\cdot)\) \(\chi_{4009}(146,\cdot)\) \(\chi_{4009}(147,\cdot)\) \(\chi_{4009}(186,\cdot)\) \(\chi_{4009}(200,\cdot)\) \(\chi_{4009}(219,\cdot)\) \(\chi_{4009}(238,\cdot)\) \(\chi_{4009}(279,\cdot)\) \(\chi_{4009}(300,\cdot)\) \(\chi_{4009}(326,\cdot)\) \(\chi_{4009}(357,\cdot)\) \(\chi_{4009}(409,\cdot)\) \(\chi_{4009}(440,\cdot)\) \(\chi_{4009}(450,\cdot)\) \(\chi_{4009}(490,\cdot)\) \(\chi_{4009}(508,\cdot)\) \(\chi_{4009}(546,\cdot)\) \(\chi_{4009}(554,\cdot)\) \(\chi_{4009}(622,\cdot)\) \(\chi_{4009}(641,\cdot)\) \(\chi_{4009}(660,\cdot)\) \(\chi_{4009}(661,\cdot)\) ...

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 630 polynomial (not computed)

Values on generators

\((2111,1901)\) → \((e\left(\frac{13}{18}\right),e\left(\frac{59}{70}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4009 }(60, a) \) \(1\)\(1\)\(e\left(\frac{178}{315}\right)\)\(e\left(\frac{199}{315}\right)\)\(e\left(\frac{41}{315}\right)\)\(e\left(\frac{256}{315}\right)\)\(e\left(\frac{62}{315}\right)\)\(e\left(\frac{103}{210}\right)\)\(e\left(\frac{73}{105}\right)\)\(e\left(\frac{83}{315}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{22}{105}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4009 }(60,a) \;\) at \(\;a = \) e.g. 2