Properties

Label 6019.68
Modulus $6019$
Conductor $6019$
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(6019, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,52]))
 
pari: [g,chi] = znchar(Mod(68,6019))
 

Basic properties

Modulus: \(6019\)
Conductor: \(6019\)
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 6019.bv

\(\chi_{6019}(68,\cdot)\) \(\chi_{6019}(94,\cdot)\) \(\chi_{6019}(315,\cdot)\) \(\chi_{6019}(373,\cdot)\) \(\chi_{6019}(692,\cdot)\) \(\chi_{6019}(1296,\cdot)\) \(\chi_{6019}(1888,\cdot)\) \(\chi_{6019}(1985,\cdot)\) \(\chi_{6019}(2447,\cdot)\) \(\chi_{6019}(2817,\cdot)\) \(\chi_{6019}(2921,\cdot)\) \(\chi_{6019}(3318,\cdot)\) \(\chi_{6019}(3363,\cdot)\) \(\chi_{6019}(3435,\cdot)\) \(\chi_{6019}(3799,\cdot)\) \(\chi_{6019}(3883,\cdot)\) \(\chi_{6019}(4624,\cdot)\) \(\chi_{6019}(4858,\cdot)\) \(\chi_{6019}(4923,\cdot)\) \(\chi_{6019}(5684,\cdot)\)

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

\((2316,1392)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{26}{33}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6019 }(68, a) \) \(1\)\(1\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{7}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6019 }(68,a) \;\) at \(\;a = \) e.g. 2