Properties

Label 5225.248
Modulus $5225$
Conductor $275$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,18,0]))
 
pari: [g,chi] = znchar(Mod(248,5225))
 

Basic properties

Modulus: \(5225\)
Conductor: \(275\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{275}(248,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5225.dm

\(\chi_{5225}(248,\cdot)\) \(\chi_{5225}(1008,\cdot)\) \(\chi_{5225}(1217,\cdot)\) \(\chi_{5225}(1388,\cdot)\) \(\chi_{5225}(1502,\cdot)\) \(\chi_{5225}(3022,\cdot)\) \(\chi_{5225}(3687,\cdot)\) \(\chi_{5225}(4903,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\((2927,2851,4676)\) → \((e\left(\frac{11}{20}\right),e\left(\frac{9}{10}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 5225 }(248, a) \) \(1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(-1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5225 }(248,a) \;\) at \(\;a = \) e.g. 2