Denote by the smallest integer .
Let tex2html_wrap_inline$S_k^(N)$ be the new subspace of tex2html_wrap_inline$S_k(N)$. It is the orthogonal complement, with respect to the Peterson pairing (VII, §5, [L]), of the subspace spanned by the images of tex2html_wrap_inline$S_k(M)$ for tex2html_wrap_inline$M|N$ under the natural inclusion maps. Let tex2html_wrap_inline$T^$ be the image of the Hecke algebra in the ring of endomorphisms of tex2html_wrap_inline$S_k^(N)$. By (VIII, §3, [L]), tex2html_wrap_inline$S_k^(N)$is a direct sum of distinct one dimensional eigenspaces. We call tex2html_wrap_inline$f&isin#in;S_k^(N)$ a newform if it is an eigenform for all Hecke operators tex2html_wrap_inline$T_p$ and if it is normalized so that tex2html_wrap_inline$a_1(f)=1$.
theorem_type[proposition][theorem][][plain][][]
If tex2html_wrap_inline$f$ is a newform level tex2html_wrap_inline$N$ and tex2html_wrap_inline$p|N$, then
displaymatha_p(f) = cases±p^k/2 -1& if $p||N$
0 & if $p^2|N$.proof
See the end of §6 in [DI].
Fix a square free positive integer tex2html_wrap_inline$N$. Let tex2html_wrap_inline${p_1,p_2,...p_s}$ be a subset (possibly empty, in which case tex2html_wrap_inline$s=0$) of the prime divisors of tex2html_wrap_inline$N$and set displaymathr:=[k&mu#mu;(N)(12·2^s)].
theorem_type[theorem][theorem][section][plain][][] Let tex2html_wrap_inline$&lambda#lambda;$ be a prime ideal in the ring of integers tex2html_wrap_inline$O$ of some number field. Suppose tex2html_wrap_inline$f$ and tex2html_wrap_inline$g$ are newforms in tex2html_wrap_inline$S_k^(N;O)$. Assume enumerate tex2html_wrap_inline$a_n(f-g) &equiv#equiv;0 &lambda#lambda;$ for tex2html_wrap_inline$n&le#leq;r$ and tex2html_wrap_inline$a_p_i(f) = b_p_i(g)$ for each tex2html_wrap_inline$i=1,...s$. Then tex2html_wrap_inline$f&equiv#equiv;g&lambda#lambda;$. proof Theorem 2 of [S]. Note that by Proposition tex2html_wrap_inline$a_p(f) = ±b_p(f)$.
I wonder: is displaymathT^=Z{ T_1,..., T_r, T_p_1,..., T_p_s}?I DON'T see that it does because theorem 3.5 says that, essentially, the first vectors of first tex2html_wrap_inline$r$ entries of a basis of eigenforms are all different. But, there's no reason I can see that they have to be linearly independent.