Properties

Label 5225.1521
Modulus $5225$
Conductor $275$
Order $5$
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(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,8,0]))
 
pari: [g,chi] = znchar(Mod(1521,5225))
 

Basic properties

Modulus: \(5225\)
Conductor: \(275\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(5\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{275}(146,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5225.q

\(\chi_{5225}(1521,\cdot)\) \(\chi_{5225}(2281,\cdot)\) \(\chi_{5225}(3991,\cdot)\) \(\chi_{5225}(4086,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 5.5.5719140625.3

Values on generators

\((2927,2851,4676)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{4}{5}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 5225 }(1521, a) \) \(1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5225 }(1521,a) \;\) at \(\;a = \) e.g. 2