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It has been found experimentally by Brown and Schnetz that the number ofpoints over ${\mathbb F}_p$ of a graph hypersurface is often related to thecoefficients of a modular form. In this paper I prove this relation for oneexample of a modular form of weight $4$ and two of weight $3$, refine thestatement and suggest a method of proving it for four more of weight $4$, anduse the one proved example to construct two new rigid Calabi-Yau threefoldsthat realize Hecke eigenforms of weight $4$ (one provably and oneconjecturally).

New realizations of modular forms in Calabi-Yau threefolds arising from $\phi^4$ theory