Properties

Label 4013.31
Modulus $4013$
Conductor $4013$
Order $2006$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4013, base_ring=CyclotomicField(2006))
 
M = H._module
 
chi = DirichletCharacter(H, M([1859]))
 
pari: [g,chi] = znchar(Mod(31,4013))
 

Basic properties

Modulus: \(4013\)
Conductor: \(4013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2006\)
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 4013.k

\(\chi_{4013}(4,\cdot)\) \(\chi_{4013}(6,\cdot)\) \(\chi_{4013}(9,\cdot)\) \(\chi_{4013}(13,\cdot)\) \(\chi_{4013}(15,\cdot)\) \(\chi_{4013}(17,\cdot)\) \(\chi_{4013}(25,\cdot)\) \(\chi_{4013}(28,\cdot)\) \(\chi_{4013}(31,\cdot)\) \(\chi_{4013}(42,\cdot)\) \(\chi_{4013}(44,\cdot)\) \(\chi_{4013}(46,\cdot)\) \(\chi_{4013}(47,\cdot)\) \(\chi_{4013}(59,\cdot)\) \(\chi_{4013}(63,\cdot)\) \(\chi_{4013}(64,\cdot)\) \(\chi_{4013}(66,\cdot)\) \(\chi_{4013}(69,\cdot)\) \(\chi_{4013}(70,\cdot)\) \(\chi_{4013}(71,\cdot)\) \(\chi_{4013}(76,\cdot)\) \(\chi_{4013}(89,\cdot)\) \(\chi_{4013}(91,\cdot)\) \(\chi_{4013}(96,\cdot)\) \(\chi_{4013}(97,\cdot)\) \(\chi_{4013}(99,\cdot)\) \(\chi_{4013}(105,\cdot)\) \(\chi_{4013}(110,\cdot)\) \(\chi_{4013}(115,\cdot)\) \(\chi_{4013}(127,\cdot)\) ...

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: $\Q(\zeta_{1003})$
Fixed field: Number field defined by a degree 2006 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{1859}{2006}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4013 }(31, a) \) \(1\)\(1\)\(e\left(\frac{1859}{2006}\right)\)\(e\left(\frac{253}{2006}\right)\)\(e\left(\frac{856}{1003}\right)\)\(e\left(\frac{1563}{2006}\right)\)\(e\left(\frac{53}{1003}\right)\)\(e\left(\frac{594}{1003}\right)\)\(e\left(\frac{1565}{2006}\right)\)\(e\left(\frac{253}{1003}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{838}{1003}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4013 }(31,a) \;\) at \(\;a = \) e.g. 2