Modular forms with prime Fourier coefficients zero - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T16:53:05Z http://mathoverflow.net/feeds/question/25461 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25461/modular-forms-with-prime-fourier-coefficients-zero Modular forms with prime Fourier coefficients zero Idoneal 2010-05-21T05:50:19Z 2010-08-12T01:46:17Z <p>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 $p$?</p> <p>If such a form cannot exist then why?</p> http://mathoverflow.net/questions/25461/modular-forms-with-prime-fourier-coefficients-zero/25495#25495 Answer by David Hansen for Modular forms with prime Fourier coefficients zero David Hansen 2010-05-21T15:21:31Z 2010-05-21T15:21:31Z <p>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</p> <p>$0=\sum_{i,j}c_i \overline{c_j} \sum_{p\leq X} \lambda_i(p)\overline{\lambda_j(p)}$. </p> <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.</p> http://mathoverflow.net/questions/25461/modular-forms-with-prime-fourier-coefficients-zero/35307#35307 Answer by Jonas Kibelbek for Modular forms with prime Fourier coefficients zero Jonas Kibelbek 2010-08-12T01:46:17Z 2010-08-12T01:46:17Z <p>It is only possible to write <em>f</em> 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<sub>2</sub>(&#8484;) ). On a noncongruence subgroup, the Hecke operators send all genuinely noncongruence forms to 0. (G. Berger, Hecke operators on noncongruence subgroups)</p>