The characterization of Eisenstein-type orthogonal modular forms

Following the notation above, let $a=k+j-3$ and $b=k-3$, with $a\equiv b\pmod 2$. Let $M_{a,b}(\operatorname{SO}(\Lambda),\vartheta_d)$ be the space of orthogonal modular forms of weight representation $W_{a,b}$ and radical character $\vartheta_d$. The subspace of forms of Eisenstein type is the span of constant functions when

$$(a,b)=(0,0)$$

that is, when $(k,j)=(3,0)$, and is trivial otherwise.

The characterization of Eisenstein-type forms. The subspace of $M_{a,b}(\operatorname{SO}(\Lambda),\vartheta_d)$ spanned by forms of Eisenstein type is the span of constant functions when $(a,b)=(0,0)$, and is trivial otherwise.

This statement identifies the forms that must be removed to isolate forms of general type in the relevant orthogonal modular-form spaces. The source presents it as the expected description following a cited proposition; its resolution is not established in the supplied text.