Properties

Label 6011.3
Modulus $6011$
Conductor $6011$
Order $601$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6011, base_ring=CyclotomicField(1202))
 
M = H._module
 
chi = DirichletCharacter(H, M([894]))
 
pari: [g,chi] = znchar(Mod(3,6011))
 

Basic properties

Modulus: \(6011\)
Conductor: \(6011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(601\)
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 6011.e

\(\chi_{6011}(3,\cdot)\) \(\chi_{6011}(9,\cdot)\) \(\chi_{6011}(20,\cdot)\) \(\chi_{6011}(27,\cdot)\) \(\chi_{6011}(46,\cdot)\) \(\chi_{6011}(58,\cdot)\) \(\chi_{6011}(60,\cdot)\) \(\chi_{6011}(81,\cdot)\) \(\chi_{6011}(112,\cdot)\) \(\chi_{6011}(136,\cdot)\) \(\chi_{6011}(138,\cdot)\) \(\chi_{6011}(154,\cdot)\) \(\chi_{6011}(174,\cdot)\) \(\chi_{6011}(180,\cdot)\) \(\chi_{6011}(182,\cdot)\) \(\chi_{6011}(187,\cdot)\) \(\chi_{6011}(205,\cdot)\) \(\chi_{6011}(221,\cdot)\) \(\chi_{6011}(236,\cdot)\) \(\chi_{6011}(243,\cdot)\) \(\chi_{6011}(245,\cdot)\) \(\chi_{6011}(248,\cdot)\) \(\chi_{6011}(251,\cdot)\) \(\chi_{6011}(278,\cdot)\) \(\chi_{6011}(284,\cdot)\) \(\chi_{6011}(292,\cdot)\) \(\chi_{6011}(301,\cdot)\) \(\chi_{6011}(326,\cdot)\) \(\chi_{6011}(336,\cdot)\) \(\chi_{6011}(337,\cdot)\) ...

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: $\Q(\zeta_{601})$
Fixed field: Number field defined by a degree 601 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{447}{601}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6011 }(3, a) \) \(1\)\(1\)\(e\left(\frac{447}{601}\right)\)\(e\left(\frac{366}{601}\right)\)\(e\left(\frac{293}{601}\right)\)\(e\left(\frac{426}{601}\right)\)\(e\left(\frac{212}{601}\right)\)\(e\left(\frac{404}{601}\right)\)\(e\left(\frac{139}{601}\right)\)\(e\left(\frac{131}{601}\right)\)\(e\left(\frac{272}{601}\right)\)\(e\left(\frac{427}{601}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6011 }(3,a) \;\) at \(\;a = \) e.g. 2