Properties

Label 85.19
Modulus 8585
Conductor 8585
Order 88
Real no
Primitive yes
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(85, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([4,7]))
 
Copy content pari:[g,chi] = znchar(Mod(19,85))
 

Basic properties

Modulus: 8585
Conductor: 8585
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 88
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
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 85.m

χ85(9,)\chi_{85}(9,\cdot) χ85(19,)\chi_{85}(19,\cdot) χ85(49,)\chi_{85}(49,\cdot) χ85(59,)\chi_{85}(59,\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(ζ8)\Q(\zeta_{8})
Fixed field: 8.8.256461670625.1

Values on generators

(52,71)(52,71)(1,e(78))(-1,e\left(\frac{7}{8}\right))

First values

aa 1-11122334466778899111112121313
χ85(19,a) \chi_{ 85 }(19, a) 1111i-ie(38)e\left(\frac{3}{8}\right)1-1e(18)e\left(\frac{1}{8}\right)e(18)e\left(\frac{1}{8}\right)iii-ie(18)e\left(\frac{1}{8}\right)e(78)e\left(\frac{7}{8}\right)11
Copy content sage:chi.jacobi_sum(n)
 
χ85(19,a)   \chi_{ 85 }(19,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ85(19,))   \tau_{ a }( \chi_{ 85 }(19,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ85(19,),χ85(n,))   J(\chi_{ 85 }(19,·),\chi_{ 85 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ85(19,))  K(a,b,\chi_{ 85 }(19,·)) \; at   a,b=\; a,b = e.g. 1,2