Let $k \ge 4$ be an even integer, and let $d$ be the dimension of the space $M_k(\operatorname{SL}_2(\mathbb{Z}))$ of modular forms of level 1 and weight $k$. Then the space of Hecke operators acting on $M_k$ also has dimension $d$. Is it spanned by $T_1, \dots, T_d$?

Equivalently (more explicitly but also more messily): if $f \in M_k(\operatorname{SL}_2(\mathbb{Z}))$ satisfies $a_i(f) = 0$ for $1 \le f \le d$, where $a_i(f)$ are the $q$-expansion coefficients of $f$, with no assumption on $a_0(f)$, then is it necessarily true that $f = 0$?

(Edit: See also this follow-up question which asks a related question for modular forms of higher level.)