$p$-th Fourier coefficients of newforms of level $\Gamma_1(N)$ with $p|N$

Question

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...

Answer 1

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, which in turn is identical to Theorem 3 in Li: Newforms and functional equations (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 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, 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 $\mathrm{GL}_2$, 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.

Answer 2

I think you can find the proof you want in:

• Andrew Ogg, On the Eigenvalues of Hecke Operators (1969)

Perhaps you can find it more thoroughly explained in Shimura's book, or on more concrete notes on Atkin-Lehner.

As for your second question, the first part is way too broad, see for example Ken Ono's "The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series".

Not so sure anymore about the number field generated by those coefficients.

Answer 3

A proof that $|a_p|=p^\frac{k-1}{2}$ in the situation you are considering is given in the proof of Theorem 3 of Winnie Li's paper Newforms and Functional Equations, where it is deduced from a result of Ogg. (She also considers the case in which the character of the newform in question is also a character mod $N/p$.)

Answer 4

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.