Magneticity is preserved by the SL2 action

Let $\Gamma(N)$ be a principal congruence subgroup, let $\mathcal{M}_k(\Gamma(N))$ be the space of modular forms of weight $k$, and let $\mathcal{M}^{\mathrm{mag}}_{k,d}(\Gamma(N))$ be its magnetic subspace of depth $d$. For $\gamma\in\mathrm{SL}_2(\mathbb{Z})$, write $f|_k\gamma$ for the weight-$k$ slash action.

SL2-action conjecture. For every $\gamma\in\mathrm{SL}_2(\mathbb{Z})$, the action

$$f\longmapsto f|_k\gamma$$

on $\mathcal{M}_k(\Gamma(N))$ restricts to the magnetic subspaces $\mathcal{M}^{\mathrm{mag}}_{k,d}(\Gamma(N))$.

Since all cusps are $\mathrm{SL}_2(\mathbb{Z})$-equivalent, this would mean that magneticity does not depend on the cusp chosen for the Fourier expansion. The conjecture is supported by computations on many examples.