摘要
Atkin and Lehner studied the theory of new forms of the space S<sub>2k</sub>（N） of cusp forms with group Γ<sub>0</sub>（N） and weight 2k and proved that S<sub>2k</sub>（N）=S<sub>2k</sub><sup>n</sup>ew（N）⊕S<sub>2k</sub><sup>o</sup>ld（N）and there exists a basis in S<sub>2k</sub><sup>n</sup>ew（N） which are eigenvectors for all Hecke operators but there exists a basis in S<sub>2k</sub><sup>o</sup>ld（N） which are eigenvectors for only those Hecke operators T（p）((p,
Atkin and Lehner[1] studied the theory of new forms of the space S2k(N) of cusp forms with group GAMMA0(N) and weight 2k and proved that S2k(N)=S2k(new)(N)+S2k(old)(N) and there exists a basis in S2k(new)(N) which are eigenvectors for all Hecke operators but there exists a basis in S2k(old)(N) which are eigenvectors for only those Hecke operators T(p)((p, N)=1). Manickam, Ramakrishnan and Vasudevan[2,3] studied the theory of new forms for half-integral and discussed the diagonalization of space S2k(q) with respect to all Hecke operators where q=3(4) is a prime. In this note we shall study the diagonalizations of the spaces M2k(q) and M(K)+1/2(q) with respect to all Hecke operators where q=3(4) is a prime and k greater-than-or-equal-to 2 is a positive integer.
