Properties

Label 6048.491
Modulus $6048$
Conductor $864$
Order $72$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6048, base_ring=CyclotomicField(72))
 
M = H._module
 
chi = DirichletCharacter(H, M([36,45,20,0]))
 
pari: [g,chi] = znchar(Mod(491,6048))
 

Basic properties

Modulus: \(6048\)
Conductor: \(864\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(72\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{864}(491,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6048.jj

\(\chi_{6048}(155,\cdot)\) \(\chi_{6048}(491,\cdot)\) \(\chi_{6048}(659,\cdot)\) \(\chi_{6048}(995,\cdot)\) \(\chi_{6048}(1163,\cdot)\) \(\chi_{6048}(1499,\cdot)\) \(\chi_{6048}(1667,\cdot)\) \(\chi_{6048}(2003,\cdot)\) \(\chi_{6048}(2171,\cdot)\) \(\chi_{6048}(2507,\cdot)\) \(\chi_{6048}(2675,\cdot)\) \(\chi_{6048}(3011,\cdot)\) \(\chi_{6048}(3179,\cdot)\) \(\chi_{6048}(3515,\cdot)\) \(\chi_{6048}(3683,\cdot)\) \(\chi_{6048}(4019,\cdot)\) \(\chi_{6048}(4187,\cdot)\) \(\chi_{6048}(4523,\cdot)\) \(\chi_{6048}(4691,\cdot)\) \(\chi_{6048}(5027,\cdot)\) \(\chi_{6048}(5195,\cdot)\) \(\chi_{6048}(5531,\cdot)\) \(\chi_{6048}(5699,\cdot)\) \(\chi_{6048}(6035,\cdot)\)

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: $\Q(\zeta_{72})$
Fixed field: Number field defined by a degree 72 polynomial

Values on generators

\((4159,3781,3809,2593)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{5}{18}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 6048 }(491, a) \) \(1\)\(1\)\(e\left(\frac{1}{72}\right)\)\(e\left(\frac{17}{72}\right)\)\(e\left(\frac{43}{72}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{11}{72}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{7}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6048 }(491,a) \;\) at \(\;a = \) e.g. 2