TITLE: Bounding the fourier coefficient field
QUESTION [14 upvotes]: Let $f = \sum_n a_n q^n \in S_2(\Gamma_0(N))$ be a normalized, non-CM, newform of weight $N \geq 1$ and level $2$. Let $K_f := {\mathbb Q}(\{a_n\}) \subset {\mathbb C}$ be the number field generated by its Fourier coefficients.
I was wondering if there is a bound known for the discriminant $\Delta_{K_f}$ of $K_f$, in terms of $N$ and (possibly) the dimension of the field $K_f$?
It is okay if this would be a terrible bound. I just need "any" bound.

REPLY [7 votes]: Here's an approach for a really bad bound.
Updated below based on comments and further reflection, but still giving a very bad bound.

First, $K_f$ is contained in the field generated by the eigenvalues of the Hecke operators $T_1$, ..., $T_m$, where $m$ is given by Sturm's bound.

Using Deligne's bounds on the size of the Hecke eigenvalues of $T_n$, one can bound the discriminant of the number field $K_n$ obtained by adjoining all eigenvalues of $T_n$ to $\mathbb Q$.

Now you can use those discriminant bounds to bound the discriminant of the compositum number field $K = K_2 K_3 \dots K_m$, which gives a bound on the discriminant of $K_f$.


This will give a bound that grows much faster than expontially in $N$.