Properties

Label 6011.21
Modulus $6011$
Conductor $6011$
Order $3005$
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(6010))
 
M = H._module
 
chi = DirichletCharacter(H, M([596]))
 
pari: [g,chi] = znchar(Mod(21,6011))
 

Basic properties

Modulus: \(6011\)
Conductor: \(6011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3005\)
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.g

\(\chi_{6011}(4,\cdot)\) \(\chi_{6011}(5,\cdot)\) \(\chi_{6011}(7,\cdot)\) \(\chi_{6011}(12,\cdot)\) \(\chi_{6011}(15,\cdot)\) \(\chi_{6011}(16,\cdot)\) \(\chi_{6011}(21,\cdot)\) \(\chi_{6011}(22,\cdot)\) \(\chi_{6011}(25,\cdot)\) \(\chi_{6011}(26,\cdot)\) \(\chi_{6011}(28,\cdot)\) \(\chi_{6011}(34,\cdot)\) \(\chi_{6011}(35,\cdot)\) \(\chi_{6011}(36,\cdot)\) \(\chi_{6011}(38,\cdot)\) \(\chi_{6011}(41,\cdot)\) \(\chi_{6011}(43,\cdot)\) \(\chi_{6011}(45,\cdot)\) \(\chi_{6011}(48,\cdot)\) \(\chi_{6011}(49,\cdot)\) \(\chi_{6011}(59,\cdot)\) \(\chi_{6011}(62,\cdot)\) \(\chi_{6011}(63,\cdot)\) \(\chi_{6011}(64,\cdot)\) \(\chi_{6011}(66,\cdot)\) \(\chi_{6011}(67,\cdot)\) \(\chi_{6011}(71,\cdot)\) \(\chi_{6011}(73,\cdot)\) \(\chi_{6011}(74,\cdot)\) \(\chi_{6011}(75,\cdot)\) ...

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 3005 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{298}{3005}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6011 }(21, a) \) \(1\)\(1\)\(e\left(\frac{298}{3005}\right)\)\(e\left(\frac{169}{601}\right)\)\(e\left(\frac{596}{3005}\right)\)\(e\left(\frac{284}{3005}\right)\)\(e\left(\frac{1143}{3005}\right)\)\(e\left(\frac{2473}{3005}\right)\)\(e\left(\frac{894}{3005}\right)\)\(e\left(\frac{338}{601}\right)\)\(e\left(\frac{582}{3005}\right)\)\(e\left(\frac{2889}{3005}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6011 }(21,a) \;\) at \(\;a = \) e.g. 2