Properties

Label 2500.1693
Modulus $2500$
Conductor $5$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2500, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3]))
 
pari: [g,chi] = znchar(Mod(1693,2500))
 

Basic properties

Modulus: \(2500\)
Conductor: \(5\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{5}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2500.f

\(\chi_{2500}(1693,\cdot)\) \(\chi_{2500}(2057,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: \(\Q(\zeta_{5})\)

Values on generators

\((1251,1877)\) → \((1,-i)\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 2500 }(1693, a) \) \(-1\)\(1\)\(i\)\(-i\)\(-1\)\(1\)\(i\)\(-i\)\(-1\)\(1\)\(i\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2500 }(1693,a) \;\) at \(\;a = \) e.g. 2