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This is a continuation of my previous papers , ,  concerning to the monstrous moonshine.The automorphism group·Oof the Leech latticeLplays an important role in the study of moonshine. Especially it is important to study theta functions associated with quadratic sublattices ofLconsisting of fixed vectors of elements of·O. In this paper, we discuss the properties that these functions are expected to satisfy in the relation to the monstrous moonshine.doi:10.1017/s0027763000001148 fatcat:3ebimhqbhzfnfmu6fzaj7qh6xe