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.



Bounding the fourier coefficient field
Here's an *approach* for a **really** **bad** bound.


**Updated below** based on comments and further reflection, but still giving a very bad bound.


1. 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.
2. 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 $\mathbb Q$.
3. 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 $N$.


