= Noncuspidal level-one Hecke eigenform
A simultaneous eigenfunction of all normalized level-one <Hecke operators> in positive even weight, with nonzero constant <Fourier coefficient>, is a multiple of the normalized <Eisenstein series>. The constant coefficient forces its $T_n$ eigenvalue to be $\sigma_{k-1}(n)$, and $a_1(T_nf)=a_n(f)$ forces $a_n(f)=a_1(f)\sigma_{k-1}(n)$. Subtracting the matching constant multiple of the <Eisenstein series> leaves a <cusp form>. Unless every coefficient vanishes, its prime-index coefficients grow like $p^{k-1}$, contradicting the <Fourier coefficient bound for a cusp form> $O(p^{k/2})$ when $k\geq4$. Weight two has no nonzero <modular forms>, by <vanishing of weight-two level-one modular forms>.
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