counting indices

Comparing relativistic and classical gravity to each other, there is the metric \(g_{\mu\nu}\) (10 entries) and the gravitational potential \(\Phi\) (1 number), the Christoffel-symbol \(\Gamma^\alpha_{\mu\nu}\) (40 entries) and the gravitational acceleration \(\partial_{i}\Phi\) (3 entries) as well as the Riemann-curvature \(R^{\alpha}_{\mu\nu\beta}\) (20 entries) and the tidal field \(\partial^2_{ij}\Phi\) (6 entries). Can you explain why the numbers of entries have to be this way?