**Integer weight case**: Modular forms in $S_{2k}(N) := S_{2k}(Gamma_{0}(N))$ can come from lower levels, and we’d like to know when that happens. This is where we call a modular form $f$ “old” if in fact it belongs to $S_{2k}(M)$ for some $M mid N$ and $M < N$. How do we characterize such forms? We notice that if $g in S_{2k}(M)$ then $g mid V(d) in S_{2k}(dM)$. One can then define the space of “oldforms” by

$$S_{2k}^{old}(N) := bigoplus_{substack{dM mid N \ M neq N}} S_{2k}(M) mid V(d).$$

The space $S_{2k}^{new}(N)$ of “newforms” will then be the orthogonal complement in $S_{2k}(N)$ of the oldform space with respect to the Petersson inner product. A “newform” is then a normalized form in $S_{2k}^{new}(N)$ that is an eigenform for all Hecke operators and for all of the Atkin-Lehner involutions $W_{p}$ for $p mid N$ and $W(N)$.

**Half-weight case**: Winfried Kohnen has built a theory of newforms for the half-weight subspaces $S_{k+1/2}^{+}(4N) := {f in S_{k+1/2}(4N) : a_{f}(n) = 0 text{ if } (-1)^{k}n equiv 2, 3 mod 4}$ where $N$ is odd and square-free. *Why must he hone in on the plus space to develop this theory?* Others such as Ueda and Manickam further develop theories of newforms on the more general space $S_{k+1/2}(4N)$, and they even go further to $S_{k+1/2}(2^{alpha}N)$ for $2 leq alpha leq 5$. But setting up the oldform space looks odd to me. For instance, for level $4$ we have

$$S_{k+1/2}^{old}(4) = S_{k+1/2}^{+}(4) oplus S_{k+1/2}(4) mid U(4).$$

In general, people like Manickam seem to like to define the oldform spaces like

$$S_{k+1/2}^{old}(4N) = sum_{substack{d mid 2N \ d < 2N}} left(S_{k+1/2}^{+}(4d) + S_{k+1/2}^{+}(4d)mid U(4N^{2}/d^{2})right) + sum_{substack{d mid 2N \ d text{ odd}}} left(S_{k+1/2}(4d) + S_{k+1/2}(4d) mid U(4N^{2}/d^{2}) right)$$

(there is an easier way to write this by defining $S_{k+1/2}(N)$ to be either plus space of level $4N$ when $N$ odd or non-plus space of level $2N$ when $N$ even).

**Why are $U$-operators used in this definition? Why can we not just define oldforms just as in the integer-weight case? Why is it difficult to get a “theory of newforms” for these half-weight spaces? What makes higher orders of $2^{alpha}$ dividing $N$ so difficult?**