Properties

Label 1248.139
Modulus $1248$
Conductor $416$
Order $24$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1248, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,15,0,16]))
 
pari: [g,chi] = znchar(Mod(139,1248))
 

Basic properties

Modulus: \(1248\)
Conductor: \(416\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{416}(139,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1248.eb

\(\chi_{1248}(139,\cdot)\) \(\chi_{1248}(211,\cdot)\) \(\chi_{1248}(451,\cdot)\) \(\chi_{1248}(523,\cdot)\) \(\chi_{1248}(763,\cdot)\) \(\chi_{1248}(835,\cdot)\) \(\chi_{1248}(1075,\cdot)\) \(\chi_{1248}(1147,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: Number field defined by a degree 24 polynomial

Values on generators

\((703,1093,833,769)\) → \((-1,e\left(\frac{5}{8}\right),1,e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1248 }(139, a) \) \(-1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{11}{12}\right)\)\(i\)\(e\left(\frac{13}{24}\right)\)\(-1\)\(e\left(\frac{17}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1248 }(139,a) \;\) at \(\;a = \) e.g. 2