Magneticity is preserved by Atkin–Lehner involutions

Let $\Gamma_1(N)$ be the congruence subgroup and let $\mathcal{M}_k(\Gamma_1(N))$ be the space of modular forms of weight $k$. Let $\mathcal{M}^{\mathrm{mag}}_{k,d}(\Gamma_1(N))$ be its magnetic subspace of depth $d$. For an exact divisor $Q$ of $N$, the Atkin–Lehner involution $W_Q$ acts on $\mathcal{M}_k(\Gamma_1(N))$.

Atkin–Lehner invariance conjecture. The action of Atkin–Lehner involutions on $\mathcal{M}_k(\Gamma_1(N))$ restricts to the magnetic subspaces

$$\mathcal{M}^{\mathrm{mag}}_{k,d}(\Gamma_1(N)).$$

The conjecture is motivated by computations of Atkin–Lehner actions on several examples of magnetic modular forms. It predicts preservation of magneticity under these involutions.