Eigenforms for $\mathcal{S}_2(\Gamma_0(88))$

I'm having trouble with the following exercise (5.8.3) from Diamond and Shurman's text on modular forms (this isn't homework for class, I just wanted to work this out on my own): $\mathcal{S}_2(\Gamma_0(11))$ has a unique normalized eigenform given by $f = \eta(\tau)^2\eta(11(\tau))^2$ where $\eta$ is the Dedekind eta function. Define $f_1(\tau) = f(2\tau), f_2(\tau) = f(4\tau), f_3(\tau) = f(8\tau)$, which we consider as elements of $\mathcal{S}_2(\Gamma_0(88))$.

I'm having trouble showing that (a) these four modular forms are linearly independent in $\mathcal{S}_2(\Gamma_0(88))$ and (b) figuring out what the action of the Hecke operators $T_p$ on each is.

All I know is that for any prime $p$, since $f$ is an eigenform, that $T_p(f) = a_p(f)f$. Moreover, for odd primes, the Hecke operator $T_p$ commutes with the level raising map and so $T_p(f_i) = a_p(f)f_i$, so the question is just to figure out the action of $T_2$ (at level 88) on the $f_i$ is. But now I'm stuck.

The hint at the end of the book suggests using a formula for the coefficients of an eigenform and the general formula giving the coefficients of $T_p(f)$. The latter formula is is the usual $a_n(T_p(f)) = a_{np}(f) + \chi(p)p^{k-1}a_{n/p}(f)$ and gives that $a_n(T_2(f)) = a_{2n}(f)$ since we're working at level 88 with the trivial Dirichlet character and so of course $\chi(2) = 0$. But I'm not sure what to do with this piece of information or if this is relevant to either parts of the exercise I'm stuck on.

• As a warmup, can you see why all four forms are eigenvalues for $T_p$ with eigenvalue $a_p(f)$ as long as $p$ is not 2 or 11? – David Loeffler Dec 7 '14 at 15:57
• Yes, away from 2 or 11, $T_p$ commutes with the level raising map $i_2: f(\tau) \mapsto f(2\tau)$ by a commutative diagram given earlier in the text (Prop 5.6.2). I was able to do the computation verifying this diagram and the one following it, which shows how level raising by $p$ commutes with $T_p$. So is it true then (by the second one of these diagrams, sorry I can't reproduce it here) that $T_2(f_1) = T_2(i_2(f)) = 2f$? But then, this $T_2$ is taken at level 22, not 88, right? – anonymous Dec 7 '14 at 18:37