Properties

Label 4013.32
Modulus $4013$
Conductor $4013$
Order $4012$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4013, base_ring=CyclotomicField(4012))
 
M = H._module
 
chi = DirichletCharacter(H, M([5]))
 
pari: [g,chi] = znchar(Mod(32,4013))
 

Basic properties

Modulus: \(4013\)
Conductor: \(4013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4012\)
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 4013.l

\(\chi_{4013}(2,\cdot)\) \(\chi_{4013}(3,\cdot)\) \(\chi_{4013}(5,\cdot)\) \(\chi_{4013}(8,\cdot)\) \(\chi_{4013}(12,\cdot)\) \(\chi_{4013}(14,\cdot)\) \(\chi_{4013}(18,\cdot)\) \(\chi_{4013}(20,\cdot)\) \(\chi_{4013}(21,\cdot)\) \(\chi_{4013}(22,\cdot)\) \(\chi_{4013}(23,\cdot)\) \(\chi_{4013}(26,\cdot)\) \(\chi_{4013}(27,\cdot)\) \(\chi_{4013}(29,\cdot)\) \(\chi_{4013}(30,\cdot)\) \(\chi_{4013}(32,\cdot)\) \(\chi_{4013}(33,\cdot)\) \(\chi_{4013}(34,\cdot)\) \(\chi_{4013}(35,\cdot)\) \(\chi_{4013}(37,\cdot)\) \(\chi_{4013}(38,\cdot)\) \(\chi_{4013}(39,\cdot)\) \(\chi_{4013}(45,\cdot)\) \(\chi_{4013}(48,\cdot)\) \(\chi_{4013}(50,\cdot)\) \(\chi_{4013}(55,\cdot)\) \(\chi_{4013}(56,\cdot)\) \(\chi_{4013}(57,\cdot)\) \(\chi_{4013}(62,\cdot)\) \(\chi_{4013}(65,\cdot)\) ...

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: $\Q(\zeta_{4012})$
Fixed field: Number field defined by a degree 4012 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{5}{4012}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4013 }(32, a) \) \(-1\)\(1\)\(e\left(\frac{5}{4012}\right)\)\(e\left(\frac{701}{4012}\right)\)\(e\left(\frac{5}{2006}\right)\)\(e\left(\frac{3181}{4012}\right)\)\(e\left(\frac{353}{2006}\right)\)\(e\left(\frac{256}{1003}\right)\)\(e\left(\frac{15}{4012}\right)\)\(e\left(\frac{701}{2006}\right)\)\(e\left(\frac{27}{34}\right)\)\(e\left(\frac{709}{1003}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4013 }(32,a) \;\) at \(\;a = \) e.g. 2