Modular forms under isogenies

Question

Let $f$ be a modular form of level 1, say an Eisenstein series to simplify. If $n$ is a natural number, what can we say about $f(n\tau)$ in terms of $f(\tau)$?

Answer 1

Here is the general picture. Let $f \in \mathcal{M}_k(N,\chi)$ be a modular form of weight $k$, level $N$, and nebentypus $\chi$. Suppose that $f$ has the Fourier expansion \[f(z) = \sum_{m = 0}^{\infty} a_f(m) e(mz).\] Then by replacing $z$ with $nz$, we see that \[f(nz) = \sum_{m = 0}^{\infty} a_f(m) e(mnz) = \sum_{m = 0}^{\infty} a_{f_n}(m) e(mz)\] with \[a_{f_n}(m) = \begin{cases} a_f\left(\frac{m}{n}\right) & \text{if $m \equiv 0 \pmod{n}$,} \\ 0 & \text{otherwise.} \end{cases}\]

Moreover, one can show that if $f \in \mathcal{M}_k(N,\chi)$ then $f(nz) \in \mathcal{M}_k(Nn,\chi)$ (this is a simple exercise). In general, one cannot say much more than this. In particular, there is not really any other relation between $f(z)$ and $f(nz)$ (they live in different spaces of modular forms).

However, the Atkin-Lehner theory of newforms and oldforms essentially states that one can decompose the space $\mathcal{S}_k(N,\chi)$ of cusp forms into a subspace spanned by newforms, which are eigenfunctions of every Hecke operator, and a subspace spanned by oldforms, which are newforms $f(z)$ in $\mathcal{S}_k(M,\chi)$ for each divisor $M$ of $N$ and the functions $f(nz)$ for each divisor $n$ of $\frac{N}{M}$.

In the adelic approach to the theory of modular forms, what this is essentially saying is that if $f$ is a newform, then $f(nz)$ lives inside the same automorphic representation as $f(z)$. This is a result of Casselman, which is essentially an adelic version of Atkin and Lehner's result.