Properties

Label 6048.3065
Modulus $6048$
Conductor $3024$
Order $36$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6048, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,34,18]))
 
pari: [g,chi] = znchar(Mod(3065,6048))
 

Basic properties

Modulus: \(6048\)
Conductor: \(3024\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3024}(2309,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6048.ij

\(\chi_{6048}(41,\cdot)\) \(\chi_{6048}(713,\cdot)\) \(\chi_{6048}(1049,\cdot)\) \(\chi_{6048}(1721,\cdot)\) \(\chi_{6048}(2057,\cdot)\) \(\chi_{6048}(2729,\cdot)\) \(\chi_{6048}(3065,\cdot)\) \(\chi_{6048}(3737,\cdot)\) \(\chi_{6048}(4073,\cdot)\) \(\chi_{6048}(4745,\cdot)\) \(\chi_{6048}(5081,\cdot)\) \(\chi_{6048}(5753,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.9008390745615103400556966960681548037381304055338896235706938737549613032765449450815488.1

Values on generators

\((4159,3781,3809,2593)\) → \((1,i,e\left(\frac{17}{18}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 6048 }(3065, a) \) \(1\)\(1\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{11}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6048 }(3065,a) \;\) at \(\;a = \) e.g. 2