Properties

Label 6013.361
Modulus $6013$
Conductor $6013$
Order $33$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,40]))
 
pari: [g,chi] = znchar(Mod(361,6013))
 

Basic properties

Modulus: \(6013\)
Conductor: \(6013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
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.bd

\(\chi_{6013}(361,\cdot)\) \(\chi_{6013}(844,\cdot)\) \(\chi_{6013}(879,\cdot)\) \(\chi_{6013}(905,\cdot)\) \(\chi_{6013}(1257,\cdot)\) \(\chi_{6013}(1523,\cdot)\) \(\chi_{6013}(2277,\cdot)\) \(\chi_{6013}(2620,\cdot)\) \(\chi_{6013}(2802,\cdot)\) \(\chi_{6013}(2977,\cdot)\) \(\chi_{6013}(3567,\cdot)\) \(\chi_{6013}(3831,\cdot)\) \(\chi_{6013}(4048,\cdot)\) \(\chi_{6013}(4239,\cdot)\) \(\chi_{6013}(4337,\cdot)\) \(\chi_{6013}(4524,\cdot)\) \(\chi_{6013}(4643,\cdot)\) \(\chi_{6013}(4841,\cdot)\) \(\chi_{6013}(5380,\cdot)\) \(\chi_{6013}(5994,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 33 polynomial

Values on generators

\((5155,3438)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{20}{33}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 6013 }(361, a) \) \(1\)\(1\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6013 }(361,a) \;\) at \(\;a = \) e.g. 2