Nonlifts in S<sub>12</sub>(K(8))<sup>−</sup>

Nonlifts in S₁₂(K(8))⁻ are old forms

The one nonlift eigenform in S₁₂(K(8))⁻ must be the old form coming from the nonlift eigenform in S₁₂(K(4))⁺. This already proves S₁₂(K(8))⁻ has at least one nonlift eigenform and hence Borcherds products are not needed to help span this Fricke eigenspace.

Nonlifts in S12(K(8))− are old forms

Nonlifts in S₁₂(K(8))⁻ are old forms