Properties

Label 5225.7
Modulus $5225$
Conductor $1045$
Order $60$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,42,20]))
 
pari: [g,chi] = znchar(Mod(7,5225))
 

Basic properties

Modulus: \(5225\)
Conductor: \(1045\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1045}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5225.gk

\(\chi_{5225}(7,\cdot)\) \(\chi_{5225}(68,\cdot)\) \(\chi_{5225}(182,\cdot)\) \(\chi_{5225}(657,\cdot)\) \(\chi_{5225}(843,\cdot)\) \(\chi_{5225}(1018,\cdot)\) \(\chi_{5225}(1432,\cdot)\) \(\chi_{5225}(1493,\cdot)\) \(\chi_{5225}(2268,\cdot)\) \(\chi_{5225}(2382,\cdot)\) \(\chi_{5225}(2857,\cdot)\) \(\chi_{5225}(3032,\cdot)\) \(\chi_{5225}(3218,\cdot)\) \(\chi_{5225}(3693,\cdot)\) \(\chi_{5225}(3868,\cdot)\) \(\chi_{5225}(4457,\cdot)\)

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

\((2927,2851,4676)\) → \((i,e\left(\frac{7}{10}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 5225 }(7, a) \) \(1\)\(1\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{11}{30}\right)\)\(i\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{13}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5225 }(7,a) \;\) at \(\;a = \) e.g. 2