Current genomic foundation models (GFMs) rely largely on extensive neural computation to implicitly infer conserved biological motifs from single-nucleotide inputs. We introduce Gengram, a conditional memory module that provides an explicit and efficient lookup primitive for multi-base motifs through a genome-specific hashing scheme. Integrated into state-of-the-art (SOTA) GFM backbones, Gengram achieves substantial improvements of up to 22.6% across functional genomics tasks and generalizes across diverse architectures. Further analysis of Gengram’s latent space reveals biologically meaningful representations aligned with established genomic principles. By introducing structured motif memory as a modeling primitive, Gengram improves empirical performance and mechanistic interpretability, offering a scalable and biology-aligned direction for next-generation GFMs. The code is available at https://github.com/zhejianglab/Gengram, and the model weights are available at https://huggingface.co/ZhejiangLab/Gengram.