Consider a classical space $M_k(N)$ of elliptic modular forms of weight $k$ for $\Gamma_0(N)$. The definition of an unramified Hecke operator $T_{p^m}$ in terms of double cosets is the disjoint union of double cosets $$\Gamma_0(N) \begin{pmatrix} a&b \\ c&d \end{pmatrix} \Gamma_0(N),$$ where $a, b, c, d \in \mathbb Z$ with $ad-bc = p^m$ and $c \equiv 0$ mod $N$. The usual definition for a ramified Hecke operator $T_{p^m}$ ($p | N$), the union of double cosets as above, now with the additional restriction that $p \nmid a$.
On the other hand, Eichler in his work on the basis problem worked with ramified Hecke operators without this restriction $p \nmid a$.
To me, the usual definition seems rather ad hoc, and the one Eichler uses seems more natural from the point of view of orders and ideals. (Note these definitions give different operators.) Of course, the usual definition is nice because it acts nicely on Fourier coefficients.
Question: Where were these ramified Hecke operators first defined?
For bonus credit: how did these definitions actually come about?
