Properties

Label 5148.5
Modulus $5148$
Conductor $1287$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5148, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,50,24,45]))
 
pari: [g,chi] = znchar(Mod(5,5148))
 

Basic properties

Modulus: \(5148\)
Conductor: \(1287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1287}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5148.ib

\(\chi_{5148}(5,\cdot)\) \(\chi_{5148}(317,\cdot)\) \(\chi_{5148}(785,\cdot)\) \(\chi_{5148}(905,\cdot)\) \(\chi_{5148}(1373,\cdot)\) \(\chi_{5148}(1721,\cdot)\) \(\chi_{5148}(1841,\cdot)\) \(\chi_{5148}(2621,\cdot)\) \(\chi_{5148}(2777,\cdot)\) \(\chi_{5148}(3089,\cdot)\) \(\chi_{5148}(3281,\cdot)\) \(\chi_{5148}(3557,\cdot)\) \(\chi_{5148}(3749,\cdot)\) \(\chi_{5148}(4217,\cdot)\) \(\chi_{5148}(4493,\cdot)\) \(\chi_{5148}(4997,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((2575,1145,937,4357)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{2}{5}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 5148 }(5, a) \) \(1\)\(1\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5148 }(5,a) \;\) at \(\;a = \) e.g. 2