Diamond Operator and Fourier Coefficients

Question

Let $d\in(\mathbb{Z}/N\mathbb{Z})^\times$, then one has the Diamond operator $\langle d \rangle$ acting on $M_k(\Gamma_1(N))$ via $\left(\begin{array}{cc}a&b\\c&d\end{array}\right)\in\Gamma_0(N)$ in the usual Hecke way.

I was wondering, given some $f\in M_k(\Gamma_1(N))$ with Fourier expansion $\sum a_nq^n$, if we know exactly how $\langle d\rangle $ acts on the Fourier coefficients $a_n$. I can't seem to find this result in the literature.

Thanks for the help!

Answer 1

Unfortunately, there's no straightforward formula for this, which is one of the reasons why one often works with the eigenspaces $M_k(\Gamma_1(N), \chi)$ instead.