Properties

Label 7524.49
Modulus $7524$
Conductor $1881$
Order $15$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7524, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,10,12,20]))
 
pari: [g,chi] = znchar(Mod(49,7524))
 

Basic properties

Modulus: \(7524\)
Conductor: \(1881\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1881}(49,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7524.dx

\(\chi_{7524}(49,\cdot)\) \(\chi_{7524}(1417,\cdot)\) \(\chi_{7524}(2401,\cdot)\) \(\chi_{7524}(3085,\cdot)\) \(\chi_{7524}(3469,\cdot)\) \(\chi_{7524}(4453,\cdot)\) \(\chi_{7524}(6505,\cdot)\) \(\chi_{7524}(6889,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 15 polynomial

Values on generators

\((3763,6689,4105,2377)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{2}{5}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 7524 }(49, a) \) \(1\)\(1\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{15}\right)\)\(1\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{4}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7524 }(49,a) \;\) at \(\;a = \) e.g. 2