Properties

Label 4027.6
Modulus $4027$
Conductor $4027$
Order $2013$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4027, base_ring=CyclotomicField(4026))
 
M = H._module
 
chi = DirichletCharacter(H, M([376]))
 
pari: [g,chi] = znchar(Mod(6,4027))
 

Basic properties

Modulus: \(4027\)
Conductor: \(4027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2013\)
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 4027.o

\(\chi_{4027}(6,\cdot)\) \(\chi_{4027}(9,\cdot)\) \(\chi_{4027}(10,\cdot)\) \(\chi_{4027}(17,\cdot)\) \(\chi_{4027}(19,\cdot)\) \(\chi_{4027}(21,\cdot)\) \(\chi_{4027}(22,\cdot)\) \(\chi_{4027}(24,\cdot)\) \(\chi_{4027}(25,\cdot)\) \(\chi_{4027}(29,\cdot)\) \(\chi_{4027}(35,\cdot)\) \(\chi_{4027}(36,\cdot)\) \(\chi_{4027}(40,\cdot)\) \(\chi_{4027}(46,\cdot)\) \(\chi_{4027}(53,\cdot)\) \(\chi_{4027}(55,\cdot)\) \(\chi_{4027}(62,\cdot)\) \(\chi_{4027}(71,\cdot)\) \(\chi_{4027}(74,\cdot)\) \(\chi_{4027}(76,\cdot)\) \(\chi_{4027}(77,\cdot)\) \(\chi_{4027}(78,\cdot)\) \(\chi_{4027}(79,\cdot)\) \(\chi_{4027}(81,\cdot)\) \(\chi_{4027}(83,\cdot)\) \(\chi_{4027}(84,\cdot)\) \(\chi_{4027}(88,\cdot)\) \(\chi_{4027}(89,\cdot)\) \(\chi_{4027}(90,\cdot)\) \(\chi_{4027}(96,\cdot)\) ...

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: $\Q(\zeta_{2013})$
Fixed field: Number field defined by a degree 2013 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{188}{2013}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4027 }(6, a) \) \(1\)\(1\)\(e\left(\frac{15}{671}\right)\)\(e\left(\frac{188}{2013}\right)\)\(e\left(\frac{30}{671}\right)\)\(e\left(\frac{784}{2013}\right)\)\(e\left(\frac{233}{2013}\right)\)\(e\left(\frac{46}{671}\right)\)\(e\left(\frac{45}{671}\right)\)\(e\left(\frac{376}{2013}\right)\)\(e\left(\frac{829}{2013}\right)\)\(e\left(\frac{1807}{2013}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4027 }(6,a) \;\) at \(\;a = \) e.g. 2