Properties

Label 832.319
Modulus 832832
Conductor 5252
Order 1212
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([6,0,11]))
 
Copy content pari:[g,chi] = znchar(Mod(319,832))
 

Basic properties

Modulus: 832832
Conductor: 5252
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 1212
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from χ52(7,)\chi_{52}(7,\cdot)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 832.bu

χ832(63,)\chi_{832}(63,\cdot) χ832(319,)\chi_{832}(319,\cdot) χ832(383,)\chi_{832}(383,\cdot) χ832(639,)\chi_{832}(639,\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(ζ12)\Q(\zeta_{12})
Fixed field: Q(ζ52)+\Q(\zeta_{52})^+

Values on generators

(703,261,769)(703,261,769)(1,1,e(1112))(-1,1,e\left(\frac{11}{12}\right))

First values

aa 1-11133557799111115151717191921212323
χ832(319,a) \chi_{ 832 }(319, a) 1111e(16)e\left(\frac{1}{6}\right)iie(712)e\left(\frac{7}{12}\right)e(13)e\left(\frac{1}{3}\right)e(1112)e\left(\frac{11}{12}\right)e(512)e\left(\frac{5}{12}\right)e(56)e\left(\frac{5}{6}\right)e(112)e\left(\frac{1}{12}\right)i-ie(23)e\left(\frac{2}{3}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ832(319,a)   \chi_{ 832 }(319,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ832(319,))   \tau_{ a }( \chi_{ 832 }(319,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ832(319,),χ832(n,))   J(\chi_{ 832 }(319,·),\chi_{ 832 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ832(319,))  K(a,b,\chi_{ 832 }(319,·)) \; at   a,b=\; a,b = e.g. 1,2