For positive integers \(k\) and \(N\), let \(g_0(k, N)\) be the dimension of the space \(S_k(\Gamma_0(N))\) of cusp forms of weight \(k\) on the subgroup \(\Gamma_0(N)\) of the modular group \(\text{SL}_2(\mathbb{Z})\), and let \(g^\#_0(k, N)\) be the dimension of the subspace of newforms in \(S_k(\Gamma_0(N))\). Up to now, \(g^\#_0(k, N)\) could only be computed recursively in terms of the dimensions \(g_0(k, d)\) for the divisors \(d\) of \(N\). The author proves an explicit formula which gives \(g_0(k, N)\) as a linear combination of rather simple multiplicative functions of \(N\). He starts from a well-known similar formula for \(g_0(k, N)\), and he uses the Atkin-Lehner theory to observe that, for every fixed \(k\), \(g^\#_0= g_0* \lambda\) is the Dirichlet convolution of \(g_0\) with the multiplicative function \(\lambda\) which satisfies \(\lambda(p)= -2\), \(\lambda(p^2)= 1\), \(\lambda(p^r)= 0\) for \(r\geq 3\), for powers of primes \(p\). Similar results are obtained for \(g^*_0(k, N)\), the number of non-isomorphic representations associated with \(S_k(\Gamma_0(N))\), and for the corresponding quantities \(g^\#_1(k, N)\) and \(g^*_1(k, N)\) which belong to the subgroup \(\Gamma_1(N)\) of the modular group. Then the author exploits the simplicity of his dimension formulas to find upper and lower bounds, precise average orders and sharp asymptotic upper and lower bounds for the four quantities under consideration. He gives a complete list of positive integers \(N\) for which \(g^\#_0(2, N)\leq 100\), and he speculates that every positive integer is taken as a value by \(g^\#_0(2, N)\). (In contrast, the set of values of \(g_0(2, N)\) has density zero.)