Can you give a non-trivial example of an integer weight cusp form which does not lie in the old subspace and it has $a\_p=0$ for all primes $p$?


If such a form cannot exist then why?



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 $p\leq X$ gives


$0=\sum\_{i,j}c\_i \overline{c\_j} \sum\_{p\leq X} \lambda\_i(p)\overline{\lambda\_j(p)}$. 


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.


