Properties

Label 5376.17
Modulus $5376$
Conductor $1344$
Order $48$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5376, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,21,24,8]))
 
pari: [g,chi] = znchar(Mod(17,5376))
 

Basic properties

Modulus: \(5376\)
Conductor: \(1344\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1344}(941,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5376.de

\(\chi_{5376}(17,\cdot)\) \(\chi_{5376}(593,\cdot)\) \(\chi_{5376}(689,\cdot)\) \(\chi_{5376}(1265,\cdot)\) \(\chi_{5376}(1361,\cdot)\) \(\chi_{5376}(1937,\cdot)\) \(\chi_{5376}(2033,\cdot)\) \(\chi_{5376}(2609,\cdot)\) \(\chi_{5376}(2705,\cdot)\) \(\chi_{5376}(3281,\cdot)\) \(\chi_{5376}(3377,\cdot)\) \(\chi_{5376}(3953,\cdot)\) \(\chi_{5376}(4049,\cdot)\) \(\chi_{5376}(4625,\cdot)\) \(\chi_{5376}(4721,\cdot)\) \(\chi_{5376}(5297,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((2815,5125,1793,4609)\) → \((1,e\left(\frac{7}{16}\right),-1,e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 5376 }(17, a) \) \(1\)\(1\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{17}{48}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{13}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5376 }(17,a) \;\) at \(\;a = \) e.g. 2