= Solution
For a <prime number> $p$, the defining finite sum for the <Hecke operator> simplifies to
$$
T_pf=p^{k-1}f(pz)+\frac1p\sum_{b=0}^{p-1}f((z+b)/p).
$$
The first summand has no $q^1$ term. In the second, summing the $p$th roots of unity selects precisely the terms whose index is divisible by $p$, and its $q^1$ coefficient is $a_p$. Comparing with $T_pf=\lambda(p)f$, using $a_1=1$, gives $\boxed{\lambda(p)=a_p}$. This also follows from the general coefficient computation in part (a).
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