A weight 2 modular form which happens to be a normalised cuspidal eigenform with rational coefficients has a natural geometric avatar---namely an elliptic curve over the rationals. It seems to be a subtle question as to how this notion generalises---this question was raised by Tony Scholl in conversation with me the other day. For example I guess I wouldn't expect a weight 3 normalised cuspidal eigenform with rational coefficients to be the $H^2$ of a smooth projective surface, because any such surface worth its salt would have (1,1)-forms coming from a hyperplane section, whereas the Hodge numbers of the motive attached to a weight 3 form are 0 and 2.

But in weight 4 one can again dream. A rigid Calabi-Yau 3-fold defined over the rationals has 2-dimensional $H^3$ and the Hodge numbers match up. Indeed there are many explicit examples of pairs $(X,f)$ with $X$ a rigid Calabi-Yau 3-fold over $\mathbf{Q}$ and $f$ a weight 4 cuspidal modular eigenform, such that the $\ell$-adic Galois representation attached to $f$ is isomorphic to $H^3(X,\mathbf{Q}_\ell)$ for all $\ell$.

The question: Is it reasonable to expect that (the motive attached to) *every* weight 4 normalised cuspidal eigenform with rational coefficients is associated to the cohomology of a rigid Calabi-Yau 3-fold over $\mathbf{Q}$?