nt.number theory - Modular forms with prime Fourier coefficients zero

Write $f=sum c_i f_i$ as a sum over new eigenforms. Your condition is thus equivalent to $sum c_i lambda_i(p)=0$ for all $p$. Taking the absolute value squared of this and summing over $pleq X$ gives

$0=sum_{i,j}c_i overline{c_j} sum_{pleq X} lambda_i(p)overline{lambda_j(p)}$.

By the pnt for Rankin-Selberg L-functions, the inner sum over primes is $sim X (log{X})^{-1}$ if $i=j$, and is $o(X (log{X})^{-1})$ otherwise. Taking $X$ very large we obtain $0=cX(log{X})^{-1}+o(X(log{X})^{-1})$, so contradiction.

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