Properties

Label 1800.1693
Modulus $1800$
Conductor $40$
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(1800, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,2,0,3]))
 
pari: [g,chi] = znchar(Mod(1693,1800))
 

Basic properties

Modulus: \(1800\)
Conductor: \(40\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{40}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1800.u

\(\chi_{1800}(757,\cdot)\) \(\chi_{1800}(1693,\cdot)\)

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: 4.0.8000.2

Values on generators

\((1351,901,1001,577)\) → \((1,-1,1,-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1800 }(1693, a) \) \(-1\)\(1\)\(-i\)\(-1\)\(-i\)\(-i\)\(1\)\(i\)\(1\)\(1\)\(i\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1800 }(1693,a) \;\) at \(\;a = \) e.g. 2