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](https://mathoverflow.net/questions/42815/how-many-hecke-operators-span-the-hecke-algebra) which asks a related question for modular forms of higher level.)



How many Hecke operators span the level 1 Hecke algebra?
The answer is yes when $k$ is a multiple of $4$. There is a unique form
of weight $k$ of the form $f\_k=1+a\_dq^d+\cdots$. When $k$ is a multiple
of $4$ this is the theta series for a putative extremal even unimodular
lattice of rank $2k$. Theorem 20 in chapter 7 of Conway and Sloane's
*Sphere Packings, Lattices and Groups* asserts that $a\_d>0$. They give
several references for the proof, including a 1969 paper of Siegel.


