Properties

Label 6025.5551
Modulus $6025$
Conductor $241$
Order $8$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6025, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,3]))
 
pari: [g,chi] = znchar(Mod(5551,6025))
 

Basic properties

Modulus: \(6025\)
Conductor: \(241\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{241}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6025.w

\(\chi_{6025}(1476,\cdot)\) \(\chi_{6025}(3826,\cdot)\) \(\chi_{6025}(5551,\cdot)\) \(\chi_{6025}(5776,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.8.47219273189051281.1

Values on generators

\((2652,2176)\) → \((1,e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 6025 }(5551, a) \) \(1\)\(1\)\(i\)\(i\)\(-1\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(e\left(\frac{5}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6025 }(5551,a) \;\) at \(\;a = \) e.g. 2