Properties

Label 7488.317
Modulus $7488$
Conductor $7488$
Order $48$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7488, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,8,36]))
 
pari: [g,chi] = znchar(Mod(317,7488))
 

Basic properties

Modulus: \(7488\)
Conductor: \(7488\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
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 7488.mk

\(\chi_{7488}(317,\cdot)\) \(\chi_{7488}(437,\cdot)\) \(\chi_{7488}(941,\cdot)\) \(\chi_{7488}(1685,\cdot)\) \(\chi_{7488}(2189,\cdot)\) \(\chi_{7488}(2309,\cdot)\) \(\chi_{7488}(2813,\cdot)\) \(\chi_{7488}(3557,\cdot)\) \(\chi_{7488}(4061,\cdot)\) \(\chi_{7488}(4181,\cdot)\) \(\chi_{7488}(4685,\cdot)\) \(\chi_{7488}(5429,\cdot)\) \(\chi_{7488}(5933,\cdot)\) \(\chi_{7488}(6053,\cdot)\) \(\chi_{7488}(6557,\cdot)\) \(\chi_{7488}(7301,\cdot)\)

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

\((703,6085,5825,5761)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{1}{6}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 7488 }(317, a) \) \(1\)\(1\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{17}{48}\right)\)\(i\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{9}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7488 }(317,a) \;\) at \(\;a = \) e.g. 2