Let $f$ be a newform of level $\Gamma\_1(N)$ and character $\chi$ which is not induced by a character mod $N/p$. I learned from [these notes](http://wstein.org/books/ribet-stein/main.pdf) by Ribet and Stein that $|a\_p|=p^{(k-1)/2}$ where $k$ is the weight of $f$. So I wonder


1, the proof of this statement,


2, is $a\_p$ for $p|N$ in the number field $K\_f$ of $f$, i.e the number field generated by all $a\_q$ for $(q,N)=1$?


Now I see that 2 is true. Why I thought it's wrong is because $p^{(k-1)/2}$ is of degree 2, which will implies in this case $2|[K\_f:\mathbb{Q}]$. Seems strange...



$p$-th Fourier coefficients of newforms of level $\Gamma_1(N)$ with $p|N$
Here is an answer to both questions.


**First question.** The quoted result is a special case of Theorem 9.1.10 in the [Ribet-Stein notes](http://wstein.org/books/ribet-stein/main.pdf), which in turn is identical to Theorem 3 in [Li: Newforms and functional equations](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0212&DMDID=DMDLOG_0047) (in fact Ribet-Stein emphasize that they follow Li's treatment closely).


As you can see, Li uses classical Atkin-Lehner theory and also refers to an older work of Ogg's from 1969. I am sure the same result could be deduced by looking at the underlying local newform at the prime $p$ in the Kirillov model, see e.g. [Schmidt's excellent notes](http://www2.math.ou.edu/~rschmidt/papers/gl2.pdf) for that purpose. (Also, I realize that Ben Linowitz answered much the same 2 hours ago.)


**Second question.** Let $S\_k^\text{new}(\Gamma\_1(N))$ denote the set of primitive newforms in $S\_k(\Gamma\_1(N))$, and note that this is a finite set. By Proposition 2.7 in [Deligne-Serre: Formes modulaires de poids 1](http://www.numdam.org/item?id=ASENS_1974_4_7_4_507_0), for any $f\in S\_k^\text{new}(\Gamma\_1(N))$ and for any field automorphism $\sigma$ of $\mathbb{C}$, there exits an $f^\sigma\in S\_k(\Gamma\_1(N))$ such that $a\_p(f^\sigma)=a\_p(f)^\sigma$. It follows that for any $f\in S\_k^\text{new}(\Gamma\_1(N))$, the $a\_p(f)$'s generate a number field $K\_f$ (of finite degree) over $\mathbb{Q}$. 


Now let $S$ be any subset of primes with Dirichlet density strictly less than $1/8$ (e.g. $S$ is finite), and consider the subfield $K\_{f,S}$ of $K\_f$ generated by the $a\_p(f)$'s with $p\not\in S$. I claim that $K\_{f,S}=K\_f$. Indeed, if $\sigma$ fixes $K\_{f,S}$, then $f^\sigma=f$ by [Ramakrishnan's version of strong multiplicity one for $\mathrm{GL}\_2$](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN356556735_0116&DMDID=DMDLOG_0040), hence $\sigma$ fixes $K\_f$ as a whole, and the claim follows by Galois theory. In particular, $K\_f$ is generated by the $a\_p(f)$'s with $p\nmid N$, and the remaining $a\_p(f)$'s with $p\mid N$ lie in this number field as well.


