How many Hecke operators span the level 1 Hecke algebra? 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.)
 A: It is known that $S_k(\operatorname{SL}_2(\mathbb{Z}))$ has a basis $(f_1,\ldots,f_d)$ satisfying $a_i(f_j) = \delta_{i,j}$ (see for example William Stein's book "Modular forms : a computational approach", Section 2.3, it is called Miller's basis). Thus the Hecke algebra of $S_k(\operatorname{SL}_2(\mathbb{Z}))$ is generated by $T_1,\ldots,T_d$.
I think one should be able to get the analogous result for $M_k(\operatorname{SL}_2(\mathbb{Z}))$ by using the fact that the Fourier coefficients of the Eisenstein series are so big with respect to cusp forms.
A: Write $k = 12\ell + k'$, where $k'$ is one of $0, 4, 6, 8, 10, 14$, and let $f_{k, m}$ be the unique weakly holomorphic modular form (poles allowed at cusps) of weight $k$ for $SL_2(\mathbb{Z})$ with Fourier expansion $f_{k, m}  = q^{-m} + \sum_{n \geq \ell+1} a_k(m, n) q^n$.  The duality of coefficients $a_k(m, n) = -a_{2-k}(n, m)$ between forms of weight $k$ and forms of weight $2-k$ holds (see http://www.math.ucla.edu/~wdduke/preprints/serre.pdf ), so the original question is equivalent to asking whether it is true that the coefficient $a_{k}(0, \ell+1)$ is never zero.  By duality, this coefficient is the negative of the constant term in $f_{2-k, \ell+1} = \frac{E_{14-k'}}{\Delta^{\ell+1}} = q^{-\ell-1} + \sum_{n=-\ell}^\infty a_{2-k}(\ell+1, n) q^n$.  Siegel's 1969 paper referenced in Robin Chapman's answer proves that this constant term is always nonzero (Theorem 2), so the answer to the original question is yes in all cases.  An English translation of Siegel's paper appears as an appendix in his Advanced Analytic Number Theory book, available online at http://www.math.tifr.res.in/~publ/ln/tifr23.pdf .
A: 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.
