The projection of $ \begin{pmatrix} a \\ 7 \end{pmatrix} $ onto $ \begin{pmatrix} -1 \\ 4 \end{pmatrix} $ is given by \[\frac{\begin{pmatrix} a \\ 7 \end{pmatrix} \cdot \begin{pmatrix} -1 \\ 4 \end{pmatrix}}{\left\| \begin{pmatrix} -1 \\ 4 \end{pmatrix} \right\|^2} \begin{pmatrix} -1 \\ 4 \end{pmatrix} = \frac{-a + 28}{17} \begin{pmatrix} -1 \\ 4 \end{pmatrix}.\]So, we want $ \frac{-a + 28}{17} = \frac{26}{17} $. Solving, we find $ a = \boxed{2} $.