I'm reading this paper (Higher Reciprocity Laws and Modular Forms of Weight One by T. Hiramatsu), and I'm trying to prove the examples given at the end. Specifically: the number of solutions of $x^3 - x - 1 \equiv 0 \pmod p$ is given by $$a(p)^2 - \left( \frac{-23}{p} \right),$$ where $(\frac{-23}{p})$ is the Legendre symbol and $a(p)$ is the $p$th coefficient of the expansion $$\eta(\tau)\eta(23\tau) = \sum_{n=1}^{\infty} a(n)q^n, \ \ \ \ \ \ \ \ q=e^{2\pi i\tau},$$ and $\eta(\tau)$ is the Dedekind eta function.
So here is what I've done. I've shown that by Euler's pentagonal number theorem $$F(\tau) = \eta(\tau)\eta(23\tau) = \sum_{u,v \in \mathbb{Z}} (-1)^{u+v}q^{((6u+1)^2 + 23(6v+1)^2)/24}.$$ From the paper, we have that F is a modular form of type $(1,\epsilon)$ of $\Gamma_0(92)$, where $$\epsilon(d) = \left( \frac{-23}{d} \right).$$
The Hecke operator acts on F by $$T_pF = F(p\tau) + \frac{1}{p}\sum_{b=0}^{p-1}F\left(\frac{\tau +b}{p}\right)$$ I don't think it should be too hard to show this is an eigenform, I probably just need to show that $a(n)$ is multiplicative and maybe some relationship between consecutive prime powers. Though, I'm unsure as to whether the multiplicativity should follow from it being a Hecke eigenform, and that I should be proving it a different way.
Once it is shown to be a Hecke eigenform, there is a theorem by Serre and Deligne, which says that there is a corresponding linear representation $$\rho: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to GL_2(\mathbb{C})$$ and if $p$ is a prime for which $\rho$ is unramified, then $$\mathrm{tr}(\rho(\text{Frob}_p)) = a(p) \quad\text{ and }\quad \det(\rho(\text{Frob}_p)) = \epsilon(p).$$ Furthermore, $\ker\rho = \mathrm{Gal}(\overline{\mathbb{Q}}/K)$, where $K$ is some finite Galois extension.
My main question is this: How do I find a polynomial $f$ such that $K$ is the splitting field of $f$? For this particular case, I know that $f(x) = x^3-x-1$ works, but I have no clue how to show this is true. My professor, sent me a link to this site, and I should be able to determine which extension it is by just looking at traces, but I'm curious as to whether there are any general methods for computing this.
I'm able to show that $a(p) = 0 \iff \rho(\text{Frob}_p)$ is a matrix of order 2 , and by looking at conjugacy classes of $S_3$ I can determine that this corresponds to the fact that $f(x)$ factors as a linear polynomial and an irreducible quadratic over $\mathbb{Z}/p\mathbb{Z}$. Similarly, $f(x)$ is irreducible if $a(p) = -1$, and $f$ splits completely if $a(p) = 2$. Showing that these are the only values $a(p)$ can take shouldn't be too hard. I imagine it just involves some modular arithmetic and that fact that the theorem by Serre and Deligne gives that $|a(p)| \leq 2$.
If I can solve all of this, the last thing I would hope to find, are conditions on $p$ for which $a(p)$ takes on the values above. I can show that $a(p) = 0$ if $\left(\frac{-23}{p}\right)=-1$. In the paper I'm reading, in a previous example, he uses the fact that if $\left(\frac{-11}{p}\right) = 1$ then there exists $a,b \in \mathbb{Z}$ such that $p = a^2 + ab + 3b^2$ to show that the coefficients of the modular form in that example are determined by whether $p = x^2 + 11y^2$ or $3p = x^2 + 11y^2$. However, that claim that $p = a^2+ab+3b^2$ relies on the fact that $\mathbb{Z}\left[\frac{1+\sqrt{-11}}{2}\right]$ is a UFD. In the example I'm trying to prove, the analogous case would be to use $p = a^2+ab+6b^2$ to split it into the cases of whether $p = x^2+23y^2$ or $6p = x^2 + 23y^2$, and I can do this with no problems. The main problem with this however, is that $\mathbb{Z}\left[\frac{1+\sqrt{-23}}{2}\right]$ has class number 3, so we don't always have that $p = a^2 + ab + 6b^2$. My final question is how can I get around this failure of unique factorization? One idea I have, is to instead use the class number and to show that $p^3 = a^2 + ab + 6b^2$ IS possible to possibly determine $a(p^3)$ and then use the recurrence relation for $a(p^n)$ to determine the value of $a(p)$. However, I'm completely stuck here and have no clue if this will actually work or will involve some kind of circular argument.
Any help would be greatly appreciated!