TITLE: $p$-th Fourier coefficients of newforms of level $\Gamma_1(N)$ with $p|N$
QUESTION [8 upvotes]: 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 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...

REPLY [2 votes]: This is really a comment, but the system won't let me post one. I think there's a confusion and that the absolute value is $p^((k-2)/2)$ where $k$ is the weight. For example there's a level $\Gamma_0 (2)$ newform of weight 8, $f=(\eta(z)\eta(2z))^8$, with $f^3 =\delta(z)\delta(2z)=q^3-24q^4+...$ Then $f=q-8q^2+...$, the $U_2$ eigenvalue is -8, and the exponent is $(8-2)/2$.
EDIT: The confusion was mine--I missed the condition on the character. My example illustrated case (iii) of Li's Theorem 3, while the question related to case (ii). I'd be grateful for an explicit example from the databases of case (ii) to further clear up my confusion.