Properties

Label 696.331
Modulus 696696
Conductor 232232
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(696, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([2,2,0,1]))
 
Copy content pari:[g,chi] = znchar(Mod(331,696))
 

Basic properties

Modulus: 696696
Conductor: 232232
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 χ232(99,)\chi_{232}(99,\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 696.v

χ696(307,)\chi_{696}(307,\cdot) χ696(331,)\chi_{696}(331,\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: 4.4.1560896.1

Values on generators

(175,349,233,553)(175,349,233,553)(1,1,1,i)(-1,-1,1,i)

First values

aa 1-111557711111313171719192323252531313535
χ696(331,a) \chi_{ 696 }(331, a) 1111111-1ii11iiii1-111i-i1-1
Copy content sage:chi.jacobi_sum(n)
 
χ696(331,a)   \chi_{ 696 }(331,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ696(331,))   \tau_{ a }( \chi_{ 696 }(331,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ696(331,),χ696(n,))   J(\chi_{ 696 }(331,·),\chi_{ 696 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ696(331,))  K(a,b,\chi_{ 696 }(331,·)) \; at   a,b=\; a,b = e.g. 1,2