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Introduction.It is well known that the orthogonal complement of the subspace of cusp forms with respect to the Petersson inner product is generated by the Eisenstein series of weight k ≥ 5/2. Moreover, in [5] it was shown that the orthogonal complement E 3/2 (4D, χ ) of the space of cusp forms of weight 3/2 with level 4D, D a square free integer, is generated by some Eisenstein series, which were explicitly constructed. Here is a positive divisor of D.In this paper we compute the dimension of the space spanned by the theta series of the genera of positive definite ternary forms of level 4D and find linear relations among them; first we find all distinct genera of positive definite ternary forms of level 4D, D square free, with character χ and find a maximal independent set of the space spanned by the genus theta series. Secondly, by checking the values of the genus theta series at all cusps of Γ 0 (4D) explicitly, linear relations among them are found. As a result we show that the Eisenstein space E 3/2 (4P, χ ) of prime level P is spanned by the theta series of the genera of positive definite ternary forms.

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