Properties

Label 4009.33
Modulus $4009$
Conductor $4009$
Order $126$
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(126))
 
M = H._module
 
chi = DirichletCharacter(H, M([49,123]))
 
pari: [g,chi] = znchar(Mod(33,4009))
 

Basic properties

Modulus: \(4009\)
Conductor: \(4009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
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.dv

\(\chi_{4009}(32,\cdot)\) \(\chi_{4009}(33,\cdot)\) \(\chi_{4009}(242,\cdot)\) \(\chi_{4009}(243,\cdot)\) \(\chi_{4009}(261,\cdot)\) \(\chi_{4009}(599,\cdot)\) \(\chi_{4009}(743,\cdot)\) \(\chi_{4009}(877,\cdot)\) \(\chi_{4009}(982,\cdot)\) \(\chi_{4009}(1086,\cdot)\) \(\chi_{4009}(1093,\cdot)\) \(\chi_{4009}(1105,\cdot)\) \(\chi_{4009}(1193,\cdot)\) \(\chi_{4009}(1503,\cdot)\) \(\chi_{4009}(1571,\cdot)\) \(\chi_{4009}(1720,\cdot)\) \(\chi_{4009}(1782,\cdot)\) \(\chi_{4009}(1845,\cdot)\) \(\chi_{4009}(1856,\cdot)\) \(\chi_{4009}(2009,\cdot)\) \(\chi_{4009}(2067,\cdot)\) \(\chi_{4009}(2670,\cdot)\) \(\chi_{4009}(2769,\cdot)\) \(\chi_{4009}(2776,\cdot)\) \(\chi_{4009}(2853,\cdot)\) \(\chi_{4009}(2920,\cdot)\) \(\chi_{4009}(2985,\cdot)\) \(\chi_{4009}(3004,\cdot)\) \(\chi_{4009}(3111,\cdot)\) \(\chi_{4009}(3131,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((2111,1901)\) → \((e\left(\frac{7}{18}\right),e\left(\frac{41}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4009 }(33, a) \) \(1\)\(1\)\(e\left(\frac{23}{63}\right)\)\(e\left(\frac{2}{63}\right)\)\(e\left(\frac{46}{63}\right)\)\(e\left(\frac{5}{63}\right)\)\(e\left(\frac{25}{63}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{4}{63}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{17}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4009 }(33,a) \;\) at \(\;a = \) e.g. 2