\documentclass[12pt]{article} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb} \usepackage{times} \usepackage[dvips, hyperindex]{hyperref} \usepackage{enumerate} \usepackage{amsfonts, latexsym} \usepackage{stmaryrd} \newtheorem{thm}{Theorem} \newtheorem{lem}{Lemma} \newtheorem{rem}{Remark} \newtheorem{prop}{Proposition} \newtheorem{defi}{Definition} \newtheorem{question}{Question} \newtheorem{conj}{Conjecture} \newtheorem{cor}{Corollary} \newtheorem{notation}{Notation} \theoremstyle{definition} \newtheorem{example}{Example} \newtheorem{sol}{Solution} \newcommand{\ZZ}{{\mathbf Z}} \newcommand{\QQ}{{\mathbf Q}} \newcommand{\RR}{{\mathbf R}} \newcommand{\CC}{{\mathbf C}} \newcommand{\FF}{{\mathbb F}} \newcommand{\GG}{{\mathbf G}} \newcommand{\NN}{{\mathbf N}} \newcommand{\OO}{{\mathcal O}} \newcommand{\TT}{{\mathbb T}} \newcommand{\cL}{{\mathcal L}} \newcommand{\fP}{{\mathfrak P}} \newcommand{\EE}{{\ZZ[\omega]}} \newcommand{\II}{{\ZZ[i]}} \newcommand{\oo}{{\omega}} \newcommand{\pp}{{\mathfrak p}} \newcommand{\gc}{{g(\chi_{\pi})}} \newcommand{\cross}{{\times}} \newcommand{\Zbar}{{\overline{\ZZ}}} \newcommand{\Qbar}{{\overline{\QQ}}} \newcommand{\Fbar}{{\overline{\FF}}} \newcommand{\cO}{\mathcal{O}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\bb}[1]{\textbf{#1}} \newcommand{\p}{{\partial}} \renewcommand{\l}{{\ell}} \DeclareMathOperator{\an}{an} \newcommand{\norm}{\operatorname{N}} \newcommand{\sgn}{\operatorname {sgn}} \newcommand{\Frob}{\operatorname {Frob}} \newcommand{\GL}{\operatorname {GL}} \newcommand{\SL}{\operatorname {SL}} \newcommand{\PGL}{\operatorname {PGL}} \newcommand{\PSL}{\operatorname {PSL}} \newcommand{\Gal}{\operatorname {Gal}} \newcommand{\End}{\operatorname {End}} \newcommand{\Qalg}{{\mathbf Q}^{\mathrm{alg}}} \newcommand{\image}{\operatorname {Im}} \newcommand{\frob}{\operatorname {Frob}} \newcommand{\charpoly}{\operatorname {charpoly}} \newcommand{\proj}{\operatorname{proj}} \newcommand{\iso}{\cong} \newcommand{\rhobar}{{\overline{\rho}}} \newcommand{\legendre}[2]{\left(\dfrac{\strut\textstyle #1}{\strut\textstyle #2}\right)} \newcommand{\cubic}[2]{\left(\dfrac{\strut\textstyle #1}{\strut\textstyle #2}\right)_{\scriptstyle 3}} \newcommand{\cubicc}[2]{\chi_{{#2}}({ #1})} \newcommand{\quarticc}[2]{\chi_{{#2}}({ #1})} \newcommand{\ndiv}{\nmid} \newcommand{\si}{\sum_{k = 1}^{\infty}} \newcommand{\vc}[1]{\overrightarrow{{#1}}} \renewcommand{\v}[1]{\ensuremath{\mathbf{#1}}} \newcommand{\mat}[4]{ \left( \begin{smallmatrix} #1 & #2 \\ #3 & #4 \end{smallmatrix} \right)} \begin{document} \title{On the generation of the coefficient field of a newform by a single Hecke eigenvalue} \author{Koopa Tak-Lun Koo\footnote{Department of Mathematics, University of Washington, Seattle, Box 354350 WA 98195, USA; e-mail: {\tt koopakoo@gmail.com}}\,\, and William Stein\footnote{Department of Mathematics, University of Washington, Seattle, Box 354350 WA 98195, USA; e-mail: {\tt wstein@math.washington.edu}}\,\, and Gabor Wiese\footnote{Institut f\"ur Experimentelle Mathematik, Universit\"at Duisburg-Essen, Ellernstra{\ss}e 29, 45326 Essen, Germany; e-mail: {\tt gabor@pratum.net}}} \maketitle \begin{abstract} Let $f$ be a non-CM newform of weight $k \ge 2$ without nontrivial inner twists. In this article we study the set of primes~$p$ such that the eigenvalue~$a_p(f)$ of the Hecke operator $T_p$ acting on~$f$ generates the field of coefficients of~$f$. We show that this set has density $1$, and prove a natural analogue for newforms having inner twists. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation. Mathematics Subject Classification (2000): 11F30 (primary); 11F11, 11F25, 11F80, 11R45 (secondary). \end{abstract} \section{Introduction} The main aim of this paper is to prove the following theorem. \begin{thm}\label{main density} Let $f$ be a newform (i.e., a new normalized cuspidal Hecke eigenform) of weight $k \ge 2$, level $N$ and Dirichlet character~$\chi$ which does not have complex multiplication (CM, see \cite[p.~48]{Rib80}). Let $E_f = \QQ(a_n(f) \,:\,(n,N)=1)$ be the field of coefficients of~$f$ and $F_f = \QQ\left(\frac{a_n(f)^2}{\chi(n)} \,:\,(n,N)=1\right)$. \begin{enumerate}[(a)] \item The set $$\left\{p \; \textnormal{prime}: \QQ\left(\frac{a_p(f)^2}{\chi(p)}\right) = F_f \right\}$$ has density $1$. \item If $f$ does not have any nontrivial inner twists, then the set $$\left\{p \; \textnormal{prime}: \QQ(a_p(f)) = E_f \right\}$$ has density $1$. \end{enumerate} \end{thm} A twist of~$f$ by a Dirichlet character~$\epsilon$ is said to be {\em inner} if there exists a (necessarily unique) field automorphism $\sigma_\epsilon: E_f \to E_f$ such that $$a_p (f \otimes \epsilon) = a_p(f) \epsilon(p) = \sigma_\epsilon (a_p(f))$$ for almost all primes~$p$. If $N$ is square free, $k=2$ and the Dirichlet character $\chi$ of $f$ is the trivial character, then there are no nontrivial inner twists of $f$. For a discussion of inner twists we refer the reader to \cite[\S3]{Rib80} and \cite[\S3]{Rib85}. In the presence of nontrivial inner twists, the conclusion of Part~(b) of the theorem never holds. To see this, we let $\epsilon$ be a nontrivial inner twist with associated field automorphism~$\sigma_\epsilon$. The set of primes~$p$ such that $\epsilon(p) = 1$ has a positive density and for any such~$p$ we have $\sigma_\epsilon(a_p(f)) = a_p(f)$. Therefore, $a_p(f) \in E_f^{\langle \sigma \rangle} \subsetneq E_f$ for a set of primes~$p$ of positive density. \medskip In the literature there are related but weaker results in the context of Maeda's conjecture, i.e., they concern the case of level~$1$ and assume that $S_k(1)$ consists of a single Galois orbit of newforms (see, e.g., \cite{JOno} and \cite{BM2003}). We now show how Part (b) of Theorem~\ref{main density} extends the principal results of these two papers. Let $f$ be a newform of level~$N$, weight~$k \ge 2$ and trivial Dirichlet character $\chi=1$ which neither has CM nor nontrivial inner twists. This is true when $N=1$. Let $\TT$ be the $\QQ$-algebra generated by all $T_n$ with $n\ge 1$ inside $\End(S_k(N,1))$ and let $\fP$ be the kernel of the $\QQ$-algebra homomorphism $\TT \xrightarrow{T_n \mapsto a_n(f)} E_f$. As $\TT$ is reduced, the map $\TT_\fP \xrightarrow{T_n \mapsto a_n(f)} E_f$ is a ring isomorphism with $\TT_\fP$ the localization of $\TT$ at~$\fP$. Non canonically $\TT_\fP$ is also isomorphic as a $\TT_\fP$-module (equivalently as an $E_f$-vector space) to its $\QQ$-linear dual, which can be identified with the localization at~$\fP$ of the $\QQ$-vector space $S_k(N,1; \QQ)$ of cusp forms in $S_k(N,1)$ with $q$-expansion in $\QQ[[q]]$. Hence, $\QQ(a_p(f)) = E_f$ precisely means that the characteristic polynomial~$P_p \in \QQ[X]$ of~$T_p$ acting on the localization at~$\fP$ of $S_k(N,1;\QQ)$ is irreducible. Part~(b) of Theorem~\ref{main density} hence shows that the set of primes~$p$ such that $P_p$ is irreducible has density~$1$. This extends Theorem 1 of~\cite{JOno} and Theorem 1.1 of~\cite{BM2003}. Both theorems restrict to the case $N=1$ and assume that there is a unique Galois orbit of newforms, i.e., a unique~$\fP$, so that no localization is needed. Theorem~1 of~\cite{JOno} says that $$\# \{p < X \, \textnormal{prime} \, : P_p \textnormal{ is irreducible in } \QQ[X]\} \gg \frac{X}{\log X}$$ and Theorem 1.1 of~\cite{BM2003} states that there is $\delta > 0$ such that $$\# \{p < X \, \textnormal{prime} \, : P_p \textnormal{ is reducible in } \QQ[X]\} \ll \frac{X}{(\log X)^{1+\delta}}.$$ \vspace{2ex} \noindent{\bf Acknowledgements.} The authors would like to thank the MSRI, where part of this research was done, for its hospitality. The first author would like to thank his advisor Ralph Greenberg for suggesting the problem. The second author acknowledges partial support from the National Science Foundation grant No.\ 0555776, and also used \cite{sage} for some calculations related to this paper. All three authors thank Jordi Quer for useful discussions. \section{Group theoretic input} \begin{lem}\label{fh} Let $q$ be a prime power and $\epsilon$ a generator of the cyclic group $\FF_q^{\cross}.$ \begin{enumerate}[(a)] \item The conjugacy classes $c$ in $\GL_2(\FF_q)$ have the following four kinds of representatives: $$ S_a = \begin{pmatrix}a & 0 \\ 0 & a \end{pmatrix}, \; \; T_{a} = \begin{pmatrix}a & 0 \\ 1 & a \end{pmatrix}, \; \; U_{a, b} = \begin{pmatrix}a & 0 \\ 0 & b \end{pmatrix}, \; \; V_{x, y} = \begin{pmatrix}x & \epsilon y \\ y & x \end{pmatrix}$$ where $a \not= b,$ and $y \not= 0.$ \item The number of elements in each of these conjugacy classes are: $1, q^2 - 1, q^2 + q,$ and $q^2 - q$, respectively. \end{enumerate} \end{lem} \begin{proof} See Fulton-Harris~\cite{FH}, page 68. \end{proof} We use the notation $[g]_G$ for the conjugacy class of~$g$ in~$G$. \begin{prop}\label{gpthy} Let $q$ be a prime power and $r$ a positive integer. Let further $R \subseteq \widetilde{R} \subseteq \FF_{q^{r}}^\times$ be subgroups. Put $\sqrt{\widetilde{R}} = \{ s \in \FF_{q^r}^\times \,:\, s^2 \in \widetilde{R}\}$. Set $$ H = \{ g \in \GL_2(\FF_{q}) \,:\, \det (g) \in R \} $$ and let $$ G \subseteq \{ g \in \GL_2(\FF_{q^r}) \,:\, \det (g) \in \widetilde{R} \} $$ be any subgroup such that $H$ is a normal subgroup of~$G$. Then the following statements hold. \begin{enumerate}[(a)] \item The group $G/(G \cap \FF_{q^r}^\times)$ (with $\FF_{q^r}^\times$ identified with scalar matrices) is either equal to $\PSL_2(\FF_q)$ or to $\PGL_2(\FF_q)$. More precisely, if we let $\{s_1,\dots,s_n\}$ be a system of representatives for $\sqrt{\widetilde{R}}/R$, then for all $g \in G$ there is $i$ such that $g \mat {s_i^{-1}} 00 {s_i^{-1}} \in G \cap \GL_2(\FF_q)$ and $\mat {s_i}00{s_i} \in G$. \item Let $g \in G$ such that $g \mat {s_i^{-1}} 00 {s_i^{-1}} \in G \cap \GL_2(\FF_q)$ and $\mat {s_i}00{s_i} \in G$. Then $$[g]_G = [g \mat {s_i^{-1}} 00 {s_i^{-1}}]_{G \cap \GL_2(\FF_q)} \mat {s_i}00{s_i}.$$ \item Let $P(X) = X^2 - aX + b \in \FF_{q^r}[X]$ be a polynomial. Then the inequality $$ \sum_C | C | \;\le\; 2 | \widetilde{R}/R | (q^2 + q)$$ holds, where the sum runs over the conjugacy classes $C$ of~$G$ with characteristic polynomial equal to~$P(X)$. \end{enumerate} \end{prop} \begin{proof} (a) The classification of the finite subgroups of $\PGL_2(\Fbar_q)$ yields that the group $G/(G \cap \FF_{q^r}^\times)$ is either $\PGL_2(\FF_{q^u})$ or $\PSL_2(\FF_{q^u})$ for some $u \mid r$. This, however, can only occur with $u=1$, as $\PSL_2(\FF_{q^u})$ is simple. The rest is only a reformulation. (b) This follows from (a), since scalar matrices are central. (c) From (b) we get the inclusion $$\bigsqcup_C C \subseteq \bigsqcup_{i=1}^n \bigsqcup_D D \mat {s_i}00{s_i},$$ where $C$ runs over the conjugacy classes of $G$ with characteristic polynomial equal to $P(X)$ and $D$ runs over the conjugacy classes of $G\cap \GL_2(\FF_q)$ with characteristic polynomial equal to $X^2 - a s_i^{-1} X + b s_i^{-2}$ (such a conjugacy class is empty if the polynomial is not in $\FF_q[X]$). The group $G\cap \GL_2(\FF_q)$ is normal in $\GL_2(\FF_q)$, as it contains $\SL_2(\FF_q)$. Hence, any conjugacy class of $\GL_2(\FF_q)$ either has an empty intersection with $G\cap \GL_2(\FF_q)$ or is a disjoint union of conjugacy classes of $G\cap \GL_2(\FF_q)$. Consequently, by Lemma~\ref{fh}, the disjoint union $\bigsqcup_D D \mat {s_i}00{s_i}$ is equal to one of \begin{enumerate}[(i)] \item $[U_{a,b}]_{\GL_2(\FF_q)} \mat {s_i}00{s_i}$, \item $[V_{x,y}]_{\GL_2(\FF_q)} \mat {s_i}00{s_i}$ or \item $[S_a]_{\GL_2(\FF_q)} \mat {s_i}00{s_i} \sqcup [T_a]_{\GL_2(\FF_q)} \mat {s_i}00{s_i}$. \end{enumerate} Still by Lemma~\ref{fh}, the first set contains $q^2 + q$, the second set $q^2-q$ and the third one $q^2$ elements. Hence, the set $\bigsqcup_C C$ contains at most $2 | \widetilde{R}/R | (q^2+q)$ elements. \end{proof} \section{Proof} The proof of Theorem~\ref{main density} relies on the following important theorem by Ribet, which, roughly speaking, says that the image of the mod~$\l$ Galois representation attached to a fixed newform is as big as it can be for almost all primes~$\l$. \begin{thm}[Ribet]\label{ribet} Let $f$ be a Hecke eigenform of weight $k \ge 2$, level~$N$ and Dirichlet character $\chi: (\ZZ/N\ZZ)^\times \to \CC^\times$. Suppose that $f$ does not have CM. Let $E_f$ and $F_f$ be as in Theorem~\ref{main density} and denote by $\cO_{E_f}$ and $\cO_{F_f}$ the corresponding rings of integers. There exists an abelian extension $K/\QQ$ such that for almost all prime numbers~$\l$ the following statement holds: \begin{quote} Let $\widetilde{\cL}$ be a prime ideal of $\cO_{E_f}$ dividing~$\l$. Put $\cL = \widetilde{\cL} \cap \cO_{F_f}$ and $\cO_{F_f}/\cL \cong \FF$. Consider the residual Galois representation $$\rhobar_{f,\widetilde{\cL}}: \Gal(\Qbar/\QQ) \to \GL_2(\cO_{E_f}/\widetilde{\cL})$$ attached to~$f$. Then the image $\rhobar_{f,\widetilde{\cL}}(\Gal(\Qbar/K))$ is equal to $$ \{g \in \GL_2(\FF) \,:\, \det(g) \in \FF_\l^{\times (k-1)} \}.$$ \end{quote} \end{thm} \begin{proof} It suffices to take Ribet~\cite[Thm.~3.1]{Rib85} mod~$\widetilde{\cL}$. Note that $F_f$ is the field~$E_f^\Gamma$. To see this, one checks immediately that $F_f \subseteq E_f^\Gamma$ with $\Gamma$ the group of the field automorphisms associated with the inner twists as in~\cite[\S 3]{Rib80}. On the other hand, let $\sigma$ be a field embedding $E_f \to \CC$ which is the identity on~$F_f$, i.e., on $\frac{a_n(f)^2}{\chi(n)}$ for all~$n$ with $(n,N)=1$. Then $\frac{\sigma(a_n(f))^2}{a_n(f)^2} = \frac{\sigma(\chi(n))}{\chi(n)}$ is a root of unity, and, thus, so is $\epsilon(n) = \frac{\sigma(a_n(f))}{a_n(f)}$. This defines a Dirichlet character~$\epsilon$ by which~$f$ has an inner twist. Hence, $\sigma \in \Gamma$ and $F_f = E_f^\Gamma$. Ribet does not say explicitly that $K/\QQ$ is abelian, but this follows since it is a composite of abelian extensions of~$K$, which are each cut out by a character. \end{proof} \begin{rem}\label{twist} The field $F_f$ defined in Theorem~\ref{main density} is invariant under twisting. More precisely, let $\epsilon$ be any Dirichlet character and consider the twisted modular form $f \otimes \epsilon$, the Dirichlet character of which is $\chi \epsilon^2$. Then the Fourier coefficients satisfy $a_n(f \otimes \epsilon) = a_n(f) \epsilon(n)$ and, thus, $\frac{a_n(f \otimes \epsilon)^2}{\chi(n) \epsilon(n)^2} = \frac{a_n(f)^2}{\chi(n)}$. \end{rem} \begin{rem}\label{innertwists} If $f$ in Theorem~\ref{main density} does not have any nontrivial inner twists, then $K = \QQ$ and $F_f = E_f$, since $F_f = E_f^\Gamma$ with $\Gamma$ the group of field automorphisms associated with the inner twists (see the proof of Theorem~\ref{ribet}). \end{rem} \begin{thm}\label{global_density} Let $f$ be a non-CM newform of weight $k \ge 2$, level $N$ and Dirichlet character~$\chi$. Let $F_f$ be as in Theorem~\ref{main density} and let $L \subset F_f$ be any proper subfield. Then the set $$\left\{p \; \textnormal{prime}: \frac{a_p(f)^2}{\chi(p)} \in L\right\}$$ has density zero. \end{thm} \begin{proof} Let $L \subsetneq F_f$ be a proper subfield and $\OO_L$ its integer ring. We define the set $$ S := \{ \cL \subset \OO_{F_f} \textnormal{ prime ideal}: [\OO_{F_f}/\cL : \OO_L / (L \cap \cL)] \ge 2 \}.$$ Notice that this set is infinite. For, if it were finite, then all but finitely many primes would split completely in the extension $F_f/L$, which is not the case by Chebotarev's density theorem. Let $\cL \in S$ be any prime, $\l$ its residue characteristic and $\widetilde{\cL}$ a prime of $\OO_{E_f}$ lying over~$\cL$. Put $\FF_q = \OO_L / (L \cap \cL)$, $\FF_{q^r} = \OO_{F_f}/\cL$ and $\FF_{q^{rs}} = \OO_{E_f}/\widetilde{\cL}$. We have $r \ge 2$. Let $W$ be the subgroup of $\FF_{q^{rs}}^\times$ consisting of the values of~$\chi$ modulo~$\widetilde{\cL}$; its size $|W|$ is less than or equal to $| (\ZZ/N\ZZ)^\times|$. Let $R = \FF_\l^{\times (k-1)}$ be the subgroup of $(k-1)$st powers of elements in the multiplicative group $\FF_\l^{\times}$ and let $\widetilde{R} = \langle R, W \rangle \subset \FF_{q^{rs}}^\times$. The size of $\widetilde{R}$ is less than or equal to $|R| \cdot |W|$. Let $H = \{g \in \GL_2(\FF_{q^r}) \,:\, \det(g) \in R \}$ and $G = \Gal(\Qbar^{\ker{\rhobar_{f,\widetilde{\cL}}}}/\QQ)$. By Galois theory, $G$ can be identified with the image of the residual representation $\rhobar_{f,\widetilde{\cL}}$, and we shall make this identification from now on. By Theorem~\ref{ribet} we have the inclusion of groups $$ H \subseteq G \subseteq \{g \in \GL_2(\FF_{q^{rs}}): \det(g) \in \widetilde{R} \}$$ with $H$ being normal in~$G$. If $C$ is a conjugacy class of~$G$, by Chebotarev's density theorem the density of $$\{p \, \textnormal{prime} : \, [\rhobar_{f,\widetilde{\cL}}(\frob_p)]_G = C\}$$ equals $|C|/|G|.$ We consider the set $$ M_\cL := \bigsqcup_C \{ p \, \textnormal{prime} : \, [\rhobar_{f,\widetilde{\cL}}(\frob_p)]_G = C\} \supseteq \left\{ p \, \textnormal{prime} : \, \overline{\left(\frac{a_p(f)^2}{\chi(p)}\right)} \in \FF_q \right\},$$ where the reduction modulo~$\cL$ of an element $x \in \cO_{F_f}$ is denoted by $\overline{x}$ and $C$ runs over the conjugacy classes of~$G$ with characteristic polynomials equal to some $X^2-aX+b \in \FF_{q^{rs}}[X]$ such that $$a^2 \in \{ t \in \FF_{q^{rs}} \, : \, \exists u \in \FF_q \; \exists w \in W : t = uw \}$$ and automatically $b \in \widetilde{R}$. The set $M_\cL$ has the density $\delta(M_\cL) = \sum_{C}^{}\frac{|C|}{|G|}$ with $C$ as before. There are at most $2q |W|^2 \cdot |R|$ such polynomials. We are now precisely in the situation to apply Prop.~\ref{gpthy}, Part~(c), which yields the inequality $$ \delta (M_\cL) \le \frac{4 |W|^3 q (q^{2r} +q^{r})}{(q^{3r} - q^r)} = O\left(\frac{1}{q^{r - 1}}\right) \le O\left(\frac{1}{q}\right), $$ where for the denominator we used $|G| \ge |H| = |R| \cdot |\SL_2(\FF_{q^r})|$. Since $q$ is unbounded for $\cL \in S$, the intersection $M := \bigcap_{\cL \in S} M_\cL$ is a set having a density and this density is~$0$. The inclusion $$ \left\{ p \, \textnormal{prime} : \, \frac{a_p(f)^2}{\chi(p)} \in L \right\} \subseteq M $$ finishes the proof. \end{proof} \begin{proof}[Proof of Theorem~\ref{main density}] To obtain (a), it suffices to apply Theorem~\ref{global_density} to each of the finitely many sub-extension of~$F_f$. (b) follows from (a) by Remark~\ref{innertwists} and the fact that $\chi$ must take values in $\{\pm 1\}$, as otherwise $E_f$ would be a CM-field and complex conjugation would give a nontrivial inner twist. \end{proof} \section{Reducibility of Hecke polynomials: questions} Motivated by a conjecture of Maeda, there has been some speculation that for every integer $k$ and prime number $p$, the characteristic polynomial of $T_p$ acting on $S_k(1)$ is irreducible. See, for example, \cite{fj}, which verifies this for all $k<2000$ and $p<2000$. The most general such speculation might be the following question: {\em if~$f$ is a non-CM newform of level $N\geq 1$ and weight $k \ge 2$ such that some $a_p(f)$ generates the field $E_f = \QQ(a_n(f) : n\geq 1)$, do all but finitely many prime-indexed Fourier coefficients $a_p(f)$ have irreducible characteristic polynomial?} The answer in general is no. An example is given by the newform in level~$63$ and weight~$2$ that has an inner twist by $\left(\frac{\cdot}{4}\right)$. Also for non-CM newforms of weight~$2$ without nontrivial inner twists such that $[E_f:\QQ]=2$, we think that the answer is likely no. Let $f\in S_k(\Gamma_0(N))$ be a newform of weight $k$ and level $N$. The {\em degree} of $f$ is the degree of the field $E_f$, and we say that~$f$ is a {\em reducible newform} if the characteristic polynomial of $a_p(f)$ is reducible for infinitely many primes~$p$. For each even weight $k\leq 12$ and degree $d=2,3,4$, we used \cite{sage} to find newforms $f$ of weight $k$ and degree $d$. For each of these forms, we computed the {\em reducible primes} $p<1000$, i.e., the primes such that the characteristic polynomial of $a_p(f)$ is reducible. The result of this computation is given in Table~\ref{counting}. Table~\ref{newforms2} contains the number of reducible primes $p<10000$ for the first $20$ newforms of degree $2$ and weight $2$. This data inspires the following question. \begin{question} If $f\in S_2(\Gamma_0(N))$ is a newform of degree $2$, is $f$ necessarily reducible? That is, are there infinitely many primes $p$ such that $a_p(f)\in \ZZ$, or equivalently, such that the characteristic polynomial of $a_p(f)$ is reducible? \end{question} Tables~\ref{newforms3}--\ref{newforms389} contain additional data about the first few newforms of given degree and weight, which may suggest other similar questions. In particular, Table~\ref{tab:million} contains data for all primes up to $10^6$ for the first degree 2 form $f$ with $L(f,1)\neq 0$, and for the first degree 2 form $g$ with $L(g,1) = 0$. We find that there are 386 primes $<10^6$ with $a_p(f) \in \ZZ$ (i.e., has reducible characteristic polynomial), and $309$ with $a_p(g)\in \ZZ$. \begin{question} If $f\in S_2(\Gamma_0(N))$ is a newform of degree $2$, can the asymptotic behaviour of the function $$ N(x) := \#\{ p \, \textnormal{prime} : \, p < x, a_p(f) \in \ZZ \} $$ be described as a function of~$x$? \end{question} The authors intend to investigate these questions in a subsequent paper. \begin{table} \begin{center} \caption{Counting Reducible Characteristic Polynomials\label{counting}} \vspace{0.5ex} \begin{tabular}{|l|l|l|l|}\hline $k$ & $d$ & $N$ & reducible $p<1000$\\\hline 2 & 2 & 23 & 13, 19, 23, 29, 43, 109, 223, 229, 271, 463, 673, 677, 883, 991\\ 2 & 3 & 41 & 17, 41\\ 2 & 4 & 47 & 47 \\\hline 4 & 2 & 11 & 11 \\ 4 & 3 & 17 & 17 \\ 4 & 4 & 23 & 23 \\\hline 6 & 2 & 7 & 7 \\ 6 & 3 & 11 & 11 \\ 6 & 4 & 17 & 17 \\\hline 8 & 2 & 5 & 5 \\ 8 & 3 & 17 & 17 \\ 8 & 4 & 11 & 11 \\\hline 10 & 2 & 5 & 5 \\ 10 & 3 & 7 & 7 \\ 10 & 4 & 13 & 13 \\\hline 12 & 2 & 5 & 5 \\ 12 & 3 & 7 & 7 \\ 12 & 4 & 21 & 3, 7 \\ \hline \end{tabular} \end{center} \end{table} \begin{table} \begin{center} \caption{First 20 Newforms of Degree 2 and Weight 2\label{newforms2}} \vspace{0.5ex} \begin{tabular}{|l|l|l|c|}\hline $k$ & $d$ & $N$ & \#\{reducible $p<10000$\}\!\\\hline 2 & 2 & 23 & 47 \\ 2 & 2 & 29 & 42 \\ 2 & 2 & 31 & 78 \\ 2 & 2 & 35 & 48 \\ 2 & 2 & 39 & 71 \\ 2 & 2 & 43 & 43 \\ 2 & 2 & 51 & 64 \\ 2 & 2 & 55 & 95 \\ 2 & 2 & 62 & 77 \\ 2 & 2 & 63 & 622 (inner twist by $\left(\frac{\cdot}{4}\right)$)\\ \hline \end{tabular}\hspace{1em}\begin{tabular}{|l|l|l|c|}\hline $k$ & $d$ & $N$ & \#\{reducible $p<10000$\}\!\\\hline 2 & 2 & 65 & 43 \\ 2 & 2 & 65 & 90 \\ 2 & 2 & 67 & 51 \\ 2 & 2 & 67 & 19 \\ 2 & 2 & 68 & 53 \\ 2 & 2 & 69 & 47 \\ 2 & 2 & 73 & 43 \\ 2 & 2 & 73 & 55 \\ 2 & 2 & 74 & 52 \\ 2 & 2 & 74 & 21 \\ \hline \end{tabular} \end{center} \end{table} \begin{table}\label{tab:million} \begin{center} \caption{Newforms 23a and 67b: values of $\psi(x) = \#\{\text{reducible }p< x\cdot 10^5\}$ \label{newforms1000000}} \vspace{0.5ex} \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline $k$ & $d$ & $N$ & $r_{\an}$ & $1$ & $2$ & $3$ &$4$ &$5$ &$6$ &$7$ &$8$ &$9$ & $10$ \\\hline 2 & 2 & $23$ & 0 & 127 & 180 & 210 & 243 & 277 & 308 & 331 & 345 & 360 & 386 \\\hline 2 & 2 & $67$ & 1 & 111 & 159 & 195 & 218 & 240 & 257 & 276 & 288 & 301 & 309 \\\hline \end{tabular} \end{center} \end{table} \begin{table} \begin{center} \caption{First 5 Newforms of Degrees 3, 4 and Weight 2\label{newforms3}} \vspace{0.5ex} \begin{tabular}{|l|l|l|l|}\hline $k$ & $d$ & $N$ & reducible $p<10000$\\ \hline 2 & 3 & 41 & 17, 41 \\ 2 & 3 & 53 & 13, 53 \\ 2 & 3 & 61 & 61, 2087 \\ 2 & 3 & 71 & 23, 31, 71, 479, \\ &&&647, 1013, 3181\\ 2 & 3 & 71 & 13, 71, 509, 3613 \\ \hline\end{tabular} \hspace{2em} \begin{tabular}{|l|l|l|l|}\hline $k$ & $d$ & $N$ & reducible $p<10000$\\ \hline 2 & 4 & 47 & 47 \\ 2 & 4 & 95 & 5, 19 \\ 2 & 4 & 97 & 97 \\ 2 & 4 & 109 & 109, 4513 \\ 2 & 4 & 111 & 3, 37 \\ &&&\\ \hline\end{tabular} \end{center} \end{table} \begin{table} \begin{center} \caption{First 5 Newforms of Degrees 2, 3 and Weight 4\label{newforms2w4}} \vspace{0.5ex} \begin{tabular}{|l|l|l|l|}\hline $k$ & $d$ & $N$ & reducible $p<1000$\\ \hline 4 & 2 & 11 & 11 \\ 4 & 2 & 13 & 13 \\ 4 & 2 & 21 & 3, 7 \\ 4 & 2 & 27 & {\tiny 3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103,}\\ &&&{\tiny 109, 127, 139, 151, 157, 163, 181, 193, 199, 211,}\\ &&&{\tiny 223, 229,241, 271, 277, 283, 307, 313, 331, 337,}\\ &&&{\tiny 349, 367, 373, 379, 397, 409, 421, 433, 439, 457,}\\ &&&{\tiny 463, 487, 499, 523, 541, 547, 571, 577, 601, 607,}\\ &&&{\tiny 613, 619, 631, 643, 661, 673, 691, 709, 727, 733,}\\ &&&{\tiny 739, 751, 757, 769, 787, 811, 823, 829, 853, 859,}\\ &&&{\tiny 877, 883, 907, 919, 937, 967, 991, 997} \\ &&&(has inner twists)\\ 4 & 2 & 29 & 29 \\ \hline \end{tabular} \begin{tabular}{|l|l|l|l|}\hline $k$ & $d$ & $N$ & reducible $p<1000$ \\\hline 4 & 3 & 17 & 17 \\ 4 & 3 & 19 & 19 \\ 4 & 3 & 35 & 5, 7 \\ 4 & 3 & 39 & 3, 13 \\ 4 & 3 & 41 & 41 \\ &&&\\&&&\\&&&\\&&&\\&&&\\&&&\\ &&&\\ &&&\\ \hline \end{tabular} \end{center} \end{table} \begin{table} \begin{center} \caption{Newforms on $\Gamma_0(389)$ of Weight $2$\label{newforms389}} \vspace{0.5ex} \begin{tabular}{|l|l|l|l|}\hline $k$ & $d$ & $N$ & reducible $p<10000$ \\\hline 2 & 1 & 389 & none (degree 1 polynomials are all irreducible) \\ 2 & 2 & 389 & {\tiny 5, 11, 59, 97, 157, 173, 223, 389, 653, 739, 859, 947, 1033, 1283, 1549, 1667, 2207, 2417, 2909, 3121, 4337,}\\ &&&{\tiny 5431, 5647, 5689, 5879, 6151, 6323, 6373, 6607, 6763, 7583, 7589, 8363, 9013, 9371, 9767} \\ 2 & 3 & 389 & 7, 13, 389, 503, 1303, 1429, 1877, 5443 \\ 2 & 6 & 389 & 19, 389\\ 2 & 20 & 389 & 389 \\ \hline \end{tabular} \end{center} \end{table} \newpage \begin{thebibliography}{X} \bibitem[BM03]{BM2003} Baba, Srinath and Murty, Ram, \emph{Irreducibility of Hecke Polynomials}, Math. 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