Consider one of the simplest non-abelian examples of modularity. Let $$\eta(6z)\eta(18z) = q\prod_{n=1}^\infty (1 - q^{6n})(1 - q^{18n}) = q - q^7 - q^{13} -q^{19} + q^{25} + 2q^{31} - q^{37} + 2q^{43} - q^{61} - q^{67} - q^{73} - q^{79} + q^{91} - q^{97} - q^{103} \dots = \sum_{n=1}^\infty a_n q^n$$

Then $a_p + 1 =$ the number of solutions of $\{x^3 - 2 = 0\}$ in $\mathbb{Z}/p\mathbb{Z}$, for prime $p > 3$.

1) What might be the best method to prove this?

2) Is it possible to prove it without knowledge of modular forms? i.e. without using the dimension / basis of the relevant space of moduli forms of some weight and level and nebentypus. For example, can it be explained using Weyl–Kac character formula etc.?

3) Unfortunately LMFDB does not yet have a list of weight $1$ modular forms and their corresponding Artin representations. Where can we find more of such weight $1$ examples?