Connections between p=x2+3y2 and Franel numbers

The Franel numbers are given by fn=∑k=0n(nk)3(n=0,1,2,…). Let p>3 be a prime. When p≡1(mod3) and p=x2+3y2 with x,y∈Z and x≡1(mod3), we show that∑k=0p−1fk2k≡∑k=0p−1fk(−4)k≡2x−p2x(modp2). We also prove that if p≡2(mod3) then∑k=0p−1fk2k≡−2∑k=0p−1fk(−4)k≡3p((p+1)/2(p+1)/6)(modp2). In addition, we propose several related conjectures for further research.