Let $p$ be an odd prime. In 2008, E. Mortenson proved van Hamme’s following conjecture: ¶ \[∑_{k=0}^{(p-1)/2}(4k+1)\Bigl({\matrix{-1/2\\k}}\Bigr)^{3}≡(-1)^{(p-1)/2}p\ \bigl(\operatorname{mod}p^{3}\bigr).\] ¶ In this paper, we show further that ¶ \begin{eqnarray*}∑_{k=0}^{p-1}(4k+1)\Bigl({\matrix{-1/2\\k}}\Bigr)^{3}&≡&∑_{k=0}^{(p-1)/2}(4k+1)\Bigl({\matrix{-1/2\\k}}\Bigr)^{3}\\[-2pt]&≡&(-1)^{(p-1)/2}p+p^{3}E_{p-3}\ \bigl(\operatorname{mod}p^{4}\bigr),\end{eqnarray*} ¶ where $E_{0},E_{1},E_{2},…$ are Euler numbers. We also prove that if $p>3$ then ¶ \begin{eqnarray*}&&∑_{k=0}^{(p-1)/2}\frac{20k+3}{(-2^{10})^{k}}\Bigl({\matrix{4k\\k,k,k,k}}\Bigr)\\&&\quad ≡(-1)^{(p-1)/2}p\bigl(2^{p-1}+2-\bigl(2^{p-1}-1\bigr)^{2}\bigr)\ \bigl(\operatorname{mod}p^{4}\bigr).\end{eqnarray*}