Properties

Label 6011.10
Modulus $6011$
Conductor $6011$
Order $6010$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6011, base_ring=CyclotomicField(6010))
 
M = H._module
 
chi = DirichletCharacter(H, M([849]))
 
pari: [g,chi] = znchar(Mod(10,6011))
 

Basic properties

Modulus: \(6011\)
Conductor: \(6011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6010\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6011.h

\(\chi_{6011}(2,\cdot)\) \(\chi_{6011}(6,\cdot)\) \(\chi_{6011}(8,\cdot)\) \(\chi_{6011}(10,\cdot)\) \(\chi_{6011}(11,\cdot)\) \(\chi_{6011}(13,\cdot)\) \(\chi_{6011}(14,\cdot)\) \(\chi_{6011}(17,\cdot)\) \(\chi_{6011}(18,\cdot)\) \(\chi_{6011}(23,\cdot)\) \(\chi_{6011}(24,\cdot)\) \(\chi_{6011}(29,\cdot)\) \(\chi_{6011}(30,\cdot)\) \(\chi_{6011}(31,\cdot)\) \(\chi_{6011}(33,\cdot)\) \(\chi_{6011}(37,\cdot)\) \(\chi_{6011}(39,\cdot)\) \(\chi_{6011}(40,\cdot)\) \(\chi_{6011}(42,\cdot)\) \(\chi_{6011}(47,\cdot)\) \(\chi_{6011}(50,\cdot)\) \(\chi_{6011}(51,\cdot)\) \(\chi_{6011}(53,\cdot)\) \(\chi_{6011}(54,\cdot)\) \(\chi_{6011}(55,\cdot)\) \(\chi_{6011}(56,\cdot)\) \(\chi_{6011}(65,\cdot)\) \(\chi_{6011}(68,\cdot)\) \(\chi_{6011}(69,\cdot)\) \(\chi_{6011}(72,\cdot)\) ...

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: $\Q(\zeta_{3005})$
Fixed field: Number field defined by a degree 6010 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{849}{6010}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6011 }(10, a) \) \(-1\)\(1\)\(e\left(\frac{849}{6010}\right)\)\(e\left(\frac{272}{601}\right)\)\(e\left(\frac{849}{3005}\right)\)\(e\left(\frac{2381}{3005}\right)\)\(e\left(\frac{3569}{6010}\right)\)\(e\left(\frac{2227}{3005}\right)\)\(e\left(\frac{2547}{6010}\right)\)\(e\left(\frac{544}{601}\right)\)\(e\left(\frac{5611}{6010}\right)\)\(e\left(\frac{5377}{6010}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6011 }(10,a) \;\) at \(\;a = \) e.g. 2