Properties

Label 720.181
Modulus 720720
Conductor 1616
Order 44
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,1,0,0]))
 
Copy content pari:[g,chi] = znchar(Mod(181,720))
 

Basic properties

Modulus: 720720
Conductor: 1616
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 44
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from χ16(5,)\chi_{16}(5,\cdot)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 720.t

χ720(181,)\chi_{720}(181,\cdot) χ720(541,)\chi_{720}(541,\cdot)

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

Related number fields

Field of values: Q(i)\mathbb{Q}(i)
Fixed field: Q(ζ16)+\Q(\zeta_{16})^+

Values on generators

(271,181,641,577)(271,181,641,577)(1,i,1,1)(1,i,1,1)

First values

aa 1-11177111113131717191923232929313137374141
χ720(181,a) \chi_{ 720 }(181, a) 11111-1iii-i11i-i1-1i-i11ii1-1
Copy content sage:chi.jacobi_sum(n)
 
χ720(181,a)   \chi_{ 720 }(181,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ720(181,))   \tau_{ a }( \chi_{ 720 }(181,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ720(181,),χ720(n,))   J(\chi_{ 720 }(181,·),\chi_{ 720 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ720(181,))  K(a,b,\chi_{ 720 }(181,·)) \; at   a,b=\; a,b = e.g. 1,2