Pseudo-normalized Hecke eigenform and its application to extremal $2$-modular lattices

It is shown that extremal $2$-modular lattices of ranks $32$ and $48$ are generated by their vectors of minimal norm. In the proof, we use certain properties of the difference of normalized Hecke eigenforms. We refer to them as the pseudo-normalized Hecke eigenform, the concept of which is introduced in this paper.