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?
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?
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.
It is only possible to write f as a sum over Hecke eigenforms, as David does, in a space of congruence modular forms (i.e., forms on a congruence subgroup of SL_{2}(ℤ) ). On a noncongruence subgroup, the Hecke operators send all genuinely noncongruence forms to 0. (G. Berger, Hecke operators on noncongruence subgroups)