Properties

Label 4009.75
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([105,97]))
 
pari: [g,chi] = znchar(Mod(75,4009))
 

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.ee

\(\chi_{4009}(75,\cdot)\) \(\chi_{4009}(246,\cdot)\) \(\chi_{4009}(303,\cdot)\) \(\chi_{4009}(341,\cdot)\) \(\chi_{4009}(360,\cdot)\) \(\chi_{4009}(398,\cdot)\) \(\chi_{4009}(588,\cdot)\) \(\chi_{4009}(778,\cdot)\) \(\chi_{4009}(797,\cdot)\) \(\chi_{4009}(835,\cdot)\) \(\chi_{4009}(873,\cdot)\) \(\chi_{4009}(892,\cdot)\) \(\chi_{4009}(1006,\cdot)\) \(\chi_{4009}(1025,\cdot)\) \(\chi_{4009}(1196,\cdot)\) \(\chi_{4009}(1215,\cdot)\) \(\chi_{4009}(1424,\cdot)\) \(\chi_{4009}(1595,\cdot)\) \(\chi_{4009}(1652,\cdot)\) \(\chi_{4009}(1690,\cdot)\) \(\chi_{4009}(1804,\cdot)\) \(\chi_{4009}(1956,\cdot)\) \(\chi_{4009}(2032,\cdot)\) \(\chi_{4009}(2051,\cdot)\) \(\chi_{4009}(2127,\cdot)\) \(\chi_{4009}(2222,\cdot)\) \(\chi_{4009}(2241,\cdot)\) \(\chi_{4009}(2317,\cdot)\) \(\chi_{4009}(2393,\cdot)\) \(\chi_{4009}(2412,\cdot)\) ...

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

\((2111,1901)\) → \((-1,e\left(\frac{97}{210}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4009 }(75, a) \) \(1\)\(1\)\(e\left(\frac{101}{105}\right)\)\(e\left(\frac{38}{105}\right)\)\(e\left(\frac{97}{105}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{34}{105}\right)\)\(e\left(\frac{43}{210}\right)\)\(e\left(\frac{31}{35}\right)\)\(e\left(\frac{76}{105}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{29}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4009 }(75,a) \;\) at \(\;a = \) e.g. 2