Properties

Label 6048.5141
Modulus $6048$
Conductor $6048$
Order $72$
Real no
Primitive yes
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([0,45,52,12]))
 
pari: [g,chi] = znchar(Mod(5141,6048))
 

Basic properties

Modulus: \(6048\)
Conductor: \(6048\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(72\)
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 6048.iy

\(\chi_{6048}(5,\cdot)\) \(\chi_{6048}(101,\cdot)\) \(\chi_{6048}(509,\cdot)\) \(\chi_{6048}(605,\cdot)\) \(\chi_{6048}(1013,\cdot)\) \(\chi_{6048}(1109,\cdot)\) \(\chi_{6048}(1517,\cdot)\) \(\chi_{6048}(1613,\cdot)\) \(\chi_{6048}(2021,\cdot)\) \(\chi_{6048}(2117,\cdot)\) \(\chi_{6048}(2525,\cdot)\) \(\chi_{6048}(2621,\cdot)\) \(\chi_{6048}(3029,\cdot)\) \(\chi_{6048}(3125,\cdot)\) \(\chi_{6048}(3533,\cdot)\) \(\chi_{6048}(3629,\cdot)\) \(\chi_{6048}(4037,\cdot)\) \(\chi_{6048}(4133,\cdot)\) \(\chi_{6048}(4541,\cdot)\) \(\chi_{6048}(4637,\cdot)\) \(\chi_{6048}(5045,\cdot)\) \(\chi_{6048}(5141,\cdot)\) \(\chi_{6048}(5549,\cdot)\) \(\chi_{6048}(5645,\cdot)\)

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

\((4159,3781,3809,2593)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{13}{18}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 6048 }(5141, a) \) \(1\)\(1\)\(e\left(\frac{5}{72}\right)\)\(e\left(\frac{13}{72}\right)\)\(e\left(\frac{47}{72}\right)\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{43}{72}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{7}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6048 }(5141,a) \;\) at \(\;a = \) e.g. 2