Properties

Label 6013.100
Modulus $6013$
Conductor $6013$
Order $39$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6013, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([26,66]))
 
pari: [g,chi] = znchar(Mod(100,6013))
 

Basic properties

Modulus: \(6013\)
Conductor: \(6013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(39\)
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 6013.bj

\(\chi_{6013}(100,\cdot)\) \(\chi_{6013}(478,\cdot)\) \(\chi_{6013}(555,\cdot)\) \(\chi_{6013}(718,\cdot)\) \(\chi_{6013}(849,\cdot)\) \(\chi_{6013}(1362,\cdot)\) \(\chi_{6013}(1383,\cdot)\) \(\chi_{6013}(1577,\cdot)\) \(\chi_{6013}(2181,\cdot)\) \(\chi_{6013}(2221,\cdot)\) \(\chi_{6013}(2242,\cdot)\) \(\chi_{6013}(2951,\cdot)\) \(\chi_{6013}(3040,\cdot)\) \(\chi_{6013}(3350,\cdot)\) \(\chi_{6013}(3560,\cdot)\) \(\chi_{6013}(3810,\cdot)\) \(\chi_{6013}(3987,\cdot)\) \(\chi_{6013}(4209,\cdot)\) \(\chi_{6013}(4419,\cdot)\) \(\chi_{6013}(4846,\cdot)\) \(\chi_{6013}(5254,\cdot)\) \(\chi_{6013}(5632,\cdot)\) \(\chi_{6013}(5709,\cdot)\) \(\chi_{6013}(6003,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 39 polynomial

Values on generators

\((5155,3438)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{11}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 6013 }(100, a) \) \(1\)\(1\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{2}{39}\right)\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{2}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6013 }(100,a) \;\) at \(\;a = \) e.g. 2