Properties

Label 4009.141
Modulus $4009$
Conductor $4009$
Order $210$
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(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,167]))
 
pari: [g,chi] = znchar(Mod(141,4009))
 

Basic properties

Modulus: \(4009\)
Conductor: \(4009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
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.ed

\(\chi_{4009}(141,\cdot)\) \(\chi_{4009}(160,\cdot)\) \(\chi_{4009}(164,\cdot)\) \(\chi_{4009}(540,\cdot)\) \(\chi_{4009}(791,\cdot)\) \(\chi_{4009}(901,\cdot)\) \(\chi_{4009}(977,\cdot)\) \(\chi_{4009}(1019,\cdot)\) \(\chi_{4009}(1057,\cdot)\) \(\chi_{4009}(1167,\cdot)\) \(\chi_{4009}(1171,\cdot)\) \(\chi_{4009}(1262,\cdot)\) \(\chi_{4009}(1338,\cdot)\) \(\chi_{4009}(1418,\cdot)\) \(\chi_{4009}(1433,\cdot)\) \(\chi_{4009}(1471,\cdot)\) \(\chi_{4009}(1494,\cdot)\) \(\chi_{4009}(1585,\cdot)\) \(\chi_{4009}(1608,\cdot)\) \(\chi_{4009}(1779,\cdot)\) \(\chi_{4009}(1794,\cdot)\) \(\chi_{4009}(2026,\cdot)\) \(\chi_{4009}(2064,\cdot)\) \(\chi_{4009}(2269,\cdot)\) \(\chi_{4009}(2406,\cdot)\) \(\chi_{4009}(2463,\cdot)\) \(\chi_{4009}(2539,\cdot)\) \(\chi_{4009}(2573,\cdot)\) \(\chi_{4009}(2934,\cdot)\) \(\chi_{4009}(2938,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((2111,1901)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{167}{210}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4009 }(141, a) \) \(1\)\(1\)\(e\left(\frac{101}{105}\right)\)\(e\left(\frac{38}{105}\right)\)\(e\left(\frac{97}{105}\right)\)\(e\left(\frac{67}{105}\right)\)\(e\left(\frac{34}{105}\right)\)\(e\left(\frac{113}{210}\right)\)\(e\left(\frac{31}{35}\right)\)\(e\left(\frac{76}{105}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{29}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4009 }(141,a) \;\) at \(\;a = \) e.g. 2