Properties

Label 4013.10
Modulus $4013$
Conductor $4013$
Order $34$
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(34))
 
M = H._module
 
chi = DirichletCharacter(H, M([19]))
 
pari: [g,chi] = znchar(Mod(10,4013))
 

Basic properties

Modulus: \(4013\)
Conductor: \(4013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(34\)
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.e

\(\chi_{4013}(10,\cdot)\) \(\chi_{4013}(530,\cdot)\) \(\chi_{4013}(831,\cdot)\) \(\chi_{4013}(1000,\cdot)\) \(\chi_{4013}(1204,\cdot)\) \(\chi_{4013}(2039,\cdot)\) \(\chi_{4013}(2726,\cdot)\) \(\chi_{4013}(2840,\cdot)\) \(\chi_{4013}(3090,\cdot)\) \(\chi_{4013}(3250,\cdot)\) \(\chi_{4013}(3617,\cdot)\) \(\chi_{4013}(3688,\cdot)\) \(\chi_{4013}(3704,\cdot)\) \(\chi_{4013}(3729,\cdot)\) \(\chi_{4013}(3913,\cdot)\) \(\chi_{4013}(3960,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: Number field defined by a degree 34 polynomial

Values on generators

\(2\) → \(e\left(\frac{19}{34}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4013 }(10, a) \) \(1\)\(1\)\(e\left(\frac{19}{34}\right)\)\(e\left(\frac{5}{34}\right)\)\(e\left(\frac{2}{17}\right)\)\(e\left(\frac{11}{34}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{23}{34}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{13}{17}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4013 }(10,a) \;\) at \(\;a = \) e.g. 2