Properties

Label 680.627
Modulus 680680
Conductor 680680
Order 88
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(680, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,4,2,3]))
 
pari: [g,chi] = znchar(Mod(627,680))
 

Basic properties

Modulus: 680680
Conductor: 680680
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 88
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 680.bw

χ680(43,)\chi_{680}(43,\cdot) χ680(427,)\chi_{680}(427,\cdot) χ680(603,)\chi_{680}(603,\cdot) χ680(627,)\chi_{680}(627,\cdot)

sage: chi.galois_orbit()
 
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: 8.8.26261675072000000.1

Values on generators

(511,341,137,241)(511,341,137,241)(1,1,i,e(38))(-1,-1,i,e\left(\frac{3}{8}\right))

First values

aa 1-1113377991111131319192121232327272929
χ680(627,a) \chi_{ 680 }(627, a) 1111e(18)e\left(\frac{1}{8}\right)e(78)e\left(\frac{7}{8}\right)iie(58)e\left(\frac{5}{8}\right)i-ii-i11e(78)e\left(\frac{7}{8}\right)e(38)e\left(\frac{3}{8}\right)e(78)e\left(\frac{7}{8}\right)
sage: chi.jacobi_sum(n)
 
χ680(627,a)   \chi_{ 680 }(627,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ680(627,))   \tau_{ a }( \chi_{ 680 }(627,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ680(627,),χ680(n,))   J(\chi_{ 680 }(627,·),\chi_{ 680 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ680(627,))  K(a,b,\chi_{ 680 }(627,·)) \; at   a,b=\; a,b = e.g. 1,2