# ****************************************************************
#          Computing Quadratic Gauss Sums
# ****************************************************************
p=7 # Pick a prime (for now)
G=DirichletGroup(p) # Group of Dirichlet characters
c=G[1]              # select a generator for characters
L=c^ZZ((p-1)/2)    # This is the only quadratic character: Legendre symbol
print "Legendre symbol:", L
print "L^2=", L^2
print " "
GS=L.gauss_sum()/p^(1/2); GS
P=(real_part(GS), imag_part(GS)); P

a = plot([],figsize=(3,3),title='Gauss Sums for p='+str(p),frame=True,axes_labels=['$x$-axis','$y$-axis'])
a+=circle((0.0,0.0),1,color='blue')
a+=point(P,color='red', size=20)
show(a)

print "Numeric values"
z=CC(GS); ord=2; print "order=",ord, " Arg(g(c))=", z
print "z^4=", z^4
k=2*pi.n()/z.argument(); print "2 Pi/Arg = ", k, " ", floor(k)

# **************************************************************
#            Gauss Sums and Computations in Cyclotomic Fields
# **************************************************************

# The Gauss sum g(c) of a character c mod p 
# belongs to the cyclotomic field of p*(p-1) roots of unity
# To compute the argument of g(c) divide g(c) by the corresponding
# primitive root.

p=7; pp=p*(p-1)*2;
L=CyclotomicField(pp); L # special case of number field defined by x^p-1
zp=L.gen(); zp

# ***********************************************************
#          Computing the quadratic Gauss sum
# ***********************************************************

G=DirichletGroup(p) # Group of Dirichlet characters
c=G[1]              # select a generator for characters
L=c^ZZ((p-1)/2)    # This is the only quadratic character: Legendre symbol
print "Legendre symbol:", L
GS=L.gauss_sum()/p^(1/2); print "Gauss sum / sqrt(p):", GS

# ***********************************************************
#          Find the argument as a power of zeta#p*(p-1)*2
#   [Note: because of sqrt(p) we look into a quadratic extension of Q(p*(p-1))]
# ***********************************************************
eps=abs(zp-1)/3 # distance bound to a cyclotomic root
root=p*(p-1)*2  # quadratic extension of Q(zeta#p(p-1))
print "Cyclotomic field of order:", root, " snap-grid eps=", eps
print "p=", p, " Quadratic GS=", CC(GS)
for j in range(root):
    z1=CC(GS-zp^j)
    # print j, z1, abs(z1)
    if abs(z1)<eps:
        n=root/j
        print "Abs(GS-z^j)=", abs(z1), " for j=",j, "GS order n=", n, " i.e. Arg(GS)=2Pi/", n