New reciprocity laws for octic residues and nonresidues

Let Z be the set of integers, and let p be a prime of the form 8k+1. Suppose q∈Z, 2∤q, p∤q, p=c2+d2=x2+2qy2, c,d,x,y∈Z and c≡1(mod4). In this paper we establish congruences for (−q)(p−1)/8(modp) and present new reciprocity laws.