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In Goldfeld's text Automorphic forms and L-functions for GL(n,R), for a fixed prime $p$ the operator $\heartsuit \colon \mathcal{L}^2(SL_2(\mathbb{Z})\backslash \mathbb{H})\to\mathcal{L}^2_{cusp}(SL_2(\mathbb{Z})\backslash \mathbb{H})$ is introduced to prove the existense of inifinitely many even cusp forms and consequently even maass forms.

Cant we just just take the collection of functions given by $h(y)\cos (2\pi jx)$ where $h(y)$ is real valued smooth function supported in $[1,2]$ and extend it to functions on $SL_2(\mathbb{Z})\backslash \mathbb{H}$. As $j$ varies through positive integers this already gives infinitely many independent even cuspidal functions. Why introduce $\heartsuit$ ?

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    $\begingroup$ For one thing, your example... although it would indeed persuade many people... has the subtle flaw that it does not make the constant term identically 0, but only 0 in the usual fundamental domain. The fully automorphic version of this function, that is, made $SL_2(\mathbb Z)$-periodic by averaging/winding-up, has rather messy constant term at heights below $y=1$. That is, "cuspform" does not mean just "vanishes in the cusp", but something a bit more complicated. $\endgroup$ – paul garrett Oct 16 '18 at 1:58
  • $\begingroup$ ah yes ! Had only the translates in my mind. Thanks. $\endgroup$ – pks Oct 16 '18 at 2:35

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