New Hadamard matrices of order 4p2 obtained from Jacobi sums of order 16

Let be a prime. Then there are integers a,b,c,d with , , p2=a2+2(b2+c2+d2), and 2ab=c2-2cd-d2. We show that there is a regular Hadamard matrix of order 4p2 provided that p=a±2b or p=a+δ12b+4δ2c+4δ1δ2d with δi=±1.