Properties

Label 825.518
Modulus $825$
Conductor $15$
Order $4$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(825, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,3,0]))
 
pari: [g,chi] = znchar(Mod(518,825))
 

Basic properties

Modulus: \(825\)
Conductor: \(15\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{15}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 825.k

\(\chi_{825}(518,\cdot)\) \(\chi_{825}(782,\cdot)\)

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: \(\Q(\zeta_{15})^+\)

Values on generators

\((551,727,376)\) → \((-1,-i,1)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\( \chi_{ 825 }(518, a) \) \(1\)\(1\)\(i\)\(-1\)\(-i\)\(-i\)\(i\)\(1\)\(1\)\(i\)\(-1\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 825 }(518,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 825 }(518,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 825 }(518,·),\chi_{ 825 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 825 }(518,·)) \;\) at \(\; a,b = \) e.g. 1,2