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 $a_1(T_{n}f)=a_n(f)$ 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 $k\ge 4$. Weight two has no nonzero modular forms, by vanishing of weight-two level-one modular forms.