Properties

Label 3600.757
Modulus $3600$
Conductor $80$
Order $4$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1,0,1]))
 
pari: [g,chi] = znchar(Mod(757,3600))
 

Basic properties

Modulus: \(3600\)
Conductor: \(80\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{80}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3600.bb

\(\chi_{3600}(757,\cdot)\) \(\chi_{3600}(1693,\cdot)\)

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: \(\Q(\sqrt{-1}) \)
Fixed field: 4.0.256000.4

Values on generators

\((3151,901,2801,577)\) → \((1,i,1,i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3600 }(757, a) \) \(-1\)\(1\)\(-i\)\(i\)\(-1\)\(i\)\(i\)\(i\)\(i\)\(1\)\(-1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3600 }(757,a) \;\) at \(\;a = \) e.g. 2