Properties

Label 6039.26
Modulus $6039$
Conductor $2013$
Order $60$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,12,41]))
 
pari: [g,chi] = znchar(Mod(26,6039))
 

Basic properties

Modulus: \(6039\)
Conductor: \(2013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2013}(26,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6039.mu

\(\chi_{6039}(26,\cdot)\) \(\chi_{6039}(566,\cdot)\) \(\chi_{6039}(665,\cdot)\) \(\chi_{6039}(917,\cdot)\) \(\chi_{6039}(1637,\cdot)\) \(\chi_{6039}(1934,\cdot)\) \(\chi_{6039}(2555,\cdot)\) \(\chi_{6039}(2654,\cdot)\) \(\chi_{6039}(3239,\cdot)\) \(\chi_{6039}(3338,\cdot)\) \(\chi_{6039}(3446,\cdot)\) \(\chi_{6039}(3545,\cdot)\) \(\chi_{6039}(3617,\cdot)\) \(\chi_{6039}(3914,\cdot)\) \(\chi_{6039}(4085,\cdot)\) \(\chi_{6039}(4976,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1343,5491,5248)\) → \((-1,e\left(\frac{1}{5}\right),e\left(\frac{41}{60}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 6039 }(26, a) \) \(1\)\(1\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6039 }(26,a) \;\) at \(\;a = \) e.g. 2