# Can a the q-expansion of a p-adic modular form be a non-constant polynomial?

Let $E_k$ be the normalized Eisenstein series of weight k and let p be an odd prime. Then

$$E_{p^m(p-1)} = 1 \mod p^{m+1},$$

and so the p-adic limit $\lim E_{p^m(p-1)} = 1$ is a p-adic modular form of weight 0. (It is even overconvergent.)

Question: Suppose f is a p-adic modular form whose q-expansion is a polynomial. Is the q-expansion of f a constant?

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