The paper is concerned with the vector space of cusp forms of positive integral weight (even integral negative dimension) on the subgroup \(\Gamma_0(m)\) of the modular group. For every proper divisor \(m'\) of \(m\), and every divisor \(d\) of \(m/m'\), the form \(F(d\tau)\) is on \(\Gamma_0(m)\) if \(F(\tau)\) is on \(\Gamma_0(m')\); the space spanned by all such forms we call “oldforms”. The “newforms” are then the orthogonal complement of the oldforms with respect to the Petersson scalar product. the newforms have a basis of forms \(F_i(\tau)\) which are eigenfunctions of all Hecke operators \(T_p\) for \((p,m)=1\); it is proved here that each \(F_i(\tau)\) is also an eigenfunction of the corresponding operators \(U_q\), and of certain non-modular involutory operators \(W_q\) in the normalizer of \(\Gamma_0(m)\), for each prime \(q\) dividing \(m\). The eigenvalues of \(U_q\) and \(W_q\) are determined, and it is shown that any two distinct newforms have an infinity of different eigenvalues for the operators \(T_p\). Various additional results are given for the case \(m=m'q^2\), where \(q\) is an odd prime, \(2,4\), or \(8\).