# A Short Note on Inadmissible Coefficients of Weight $2$ and $2k+1$ Newforms

Malik Amir and Andreas Hatziiliou  Department of Mathematics, École
Polytechnique Fédérale de Lausanne<EMAIL_ADDRESS>Department of
Mathematics and Statistics, McGill University
<EMAIL_ADDRESS>
###### Abstract.

Soit $f(z)=q+\sum_{n\geq 2}a(n)q^{n}$ une forme de Hecke normalisée de poids
$k$ à coefficients entiers possédant une représentation galoisienne modulo 2
triviale. On généralise les résultats de Amir et Hong présentés dans [2] pour
le poids $k=2$ en éliminant ou en localisant toutes les valeurs impaires
$|\ell|<100$ des coéfficient de Fourier $a(n)$ pour $n$ respectant certaines
congruences. On étudie aussi le cas des poids impairs $k\geq 1$ de formes de
Hecke dont le caractère est donné par un caractère quadratique de Dirichlet.

Let $f(z)=q+\sum_{n\geq 2}a(n)q^{n}$ be a weight $k$ normalized newform with
integer coefficients and trivial residual mod 2 Galois representation. We
extend the results of Amir and Hong in [2] for $k=2$ by ruling out or locating
all odd prime values $|\ell|<100$ of their Fourier coefficients $a(n)$ when
$n$ satisfies some congruences. We also study the case of odd weights $k\geq
1$ newforms where the nebentypus is given by a quadratic Dirichlet character.

###### Key words and phrases:

Lucas Sequences, Lehmer’s Conjecture, Modular Forms, $L$-Functions, Elliptic
Curves

> It’s not the violence of the few that scares me, it’s the silence of the
> many.
>
> Palestine will be free

##  1\. Introduction and Statement of the Results

In an article entitled “On certain arithmetical functions” [17], Ramanujan
introduced the $\tau$-function in 1916, known as the Fourier coefficients of
the weight $12$ modular form

$\Delta(z)=q\prod_{n=1}^{\infty}(1-q^{n})^{24}:=\sum_{n=1}^{\infty}\tau(n)q^{n}=q-24q^{2}+252q^{3}-1472q^{4}+4830q^{5}-...$

where throughout $q:=e^{2\pi iz}$. It was conjectured by Ramanujan that the
$\tau$-function is multiplicative and this offered a glimpse into a much more
general theory known today as the theory of Hecke operators. Despite its
importance in the large web of mathematics and physics, basic properties of
$\tau(n)$ are still unknown. The most famous example is Lehmer’s conjecture
about the nonvanishing of $\tau(n)$. Lehmer proved that if $\tau(n)=0$, then
$n$ must be a prime [12]. One may be interested in studying odd values taken
by $\tau(n)$ or, more generally, the coefficients of any newform. This is the
question we consider as a variation of Lehmer’s original speculation.

For an odd number $\alpha$, Murty, Murty and Shorey [14] proved using linear
forms in logarithms that $\tau(n)\neq\alpha$ for all $n$ sufficiently large.
However, the bounds that they obtained are huge and computationally
impractical. Recently, using the theory of Lucas sequences, Balakrishnan,
Craig, Ono and Tsai proved in [3] and [4], together with work of Dembner and
Jain in [10], that $\tau(n)=\ell$ has no solution for $|\ell|<100$ an odd
prime. In addition, Hanada and Madhukara proved in [11] that $\tau(n)=\alpha$
has no solution for $|\alpha|<100$ an odd integer. Following these ideas, Amir
and Hong investigated weight $2$ and $3$ newforms corresponding to modular
elliptic curves and a special family of $K3$ surfaces in [2].

In this paper, we extend slightly the results of [2] on inadmissible
coefficients for $L$-functions of modular elliptic curves and give a procedure
to rule out odd prime values $\ell$, positive or negative, as coefficients of
any normalized newform of odd weight $k\geq 1$ with integer coefficients
having trivial residual mod $2$ Galois representation and a quadratic
Dirichlet character. For the rest of this paper, whenever we say newform of
weight $k$, we talk about a newform with the aforementioned properties. In the
case of weight $2$ newforms, we have the following results.

######  Theorem 1.1.

Suppose $f(z)=q+\sum_{n\geq 2}a(n)q^{n}\in
S_{2}^{new}(\Gamma_{0}(N))\cap\mathbb{Z}[[q]]$ has trivial residual mod 2
Galois representation, namely, $E/\mathbb{Q}$ is an elliptic curve of
conductor $N$ with a rational $2$-torsion point. Then the following are true.

  1. If $E/\mathbb{Q}$ has a rational $3$-torsion point, then for $n>1$ and $\gcd(n,2\cdot 3\cdot N)=1$, we have

     * 1.

If $a(n)=7,13,19,31,37$, then $n=p^{2}$ and $p=2\ (\mathrm{mod}\ 3)$.

     * 2.

If $a(n)=29$ then $n=p^{d-1}=13^{4}$ and $a(p)=\pm 2$.

     * 3.

If $a(n)=41$ then $n=p^{d-1}=43^{4}$ and $a(p)=\pm 4$.

     * 4.

If $a(n)=-19$ then $n=p^{d-1}=7^{4}$ and $a(p)=\pm 2$.

     * 5.

If $a(n)=-31$ then $n=p^{d-1}=7^{4}$ and $a(p)=\pm 4$.

     * 6.

If $a(n)=-79$ then $n=p^{d-1}=167^{4}$ and $a(p)=\pm 8$.

Furthermore,

$a(n)\not\in\\{-1,1,5,-7,11,-13,17,23,-37,-43,47,53,59,-61,-67,71,-73,83,89,-97\\}.$

  2. If $E/\mathbb{Q}$ has a rational $5$-torsion point, then for $n>1$ and $\gcd(n,2\cdot 5\cdot N)=1$, we have

     * 1.

If $\ell\equiv 1\ (\mathrm{mod}\ 5)$ and $a(n)=\ell$, then $n=p^{2}$ and
$p\equiv 4\ (\mathrm{mod}\ 5)$.

     * 2.

If $\ell\equiv 2\ (\mathrm{mod}\ 5)$, $\ell\neq-3$ and $a(n)=\ell$, then
$n=p^{2}$ and $p\equiv 2\ (\mathrm{mod}\ 5)$.

     * 3.

If $\ell\equiv 3\ (\mathrm{mod}\ 5)$, $\ell\neq 3$ and $a(n)=\ell$, then
$n=p^{2}$ and $p\equiv 1,3\ (\mathrm{mod}\ 5)$.

Furthermore,

$a(n)\not\in\\{-1,1,-11,19,29,-31,-41,59,-61,-71,79,89,-691\\}.$

######  Theorem 1.2.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ with a $2$ and
$3$-torsion point. Let $n>1$ and $gcd(n,2\cdot 3\cdot N)=1$. If $\ell\equiv 2\
(\mathrm{mod}\ 3),\ell\neq 5$ and the odd prime divisors $d$ of
$|\ell|(|\ell|-1)(|\ell|+1)$ are not congruent to $2\ (\mathrm{mod}\ 3)$, then
$a(n)\neq\ell$.

######  Theorem 1.3.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ with a $2$ and
$5$-torsion point. Let $n>1$ and $\gcd(n,2\cdot 5\cdot N)=1$.

  * 1.

If $\ell\equiv 1\ (\mathrm{mod}\ 5)$ and the odd prime divisors $d$ of
$|\ell|(|\ell|-1)(|\ell|+1)$ are not congruent to $1,3\ (\mathrm{mod}\ 5)$,
then $a(n)\neq\ell$.

  * 2.

If $\ell\equiv 2\ (\mathrm{mod}\ 5),\ell\neq-3$ and the odd prime divisors $d$
of $|\ell|(|\ell|-1)(|\ell|+1)$ are not congruent to $2,3\ (\mathrm{mod}\ 5)$,
then $a(n)\neq\ell$.

  * 3.

If $\ell\equiv 3\ (\mathrm{mod}\ 5),\ell\neq 3$ and the odd prime divisors $d$
of $|\ell|(|\ell|-1)(|\ell|+1)$ are not congruent to $2,3\ (\mathrm{mod}\ 5)$,
then $a(n)\neq\ell$.

  * 4.

If $\ell\equiv 4\ (\mathrm{mod}\ 5)$ and the odd prime divisors $d$ of
$|\ell|(|\ell|-1)(|\ell|+1)$ are not congruent to $2,4\ (\mathrm{mod}\ 5)$,
then $a(n)\neq\ell$.

######  Theorem 1.4.

Let $E/\mathbb{Q}$ be an elliptic curve with conductor $N$ and $f$ the
corresponding newform with Fourier coefficients $a(n)$. For $r=3,5,$ suppose
that $2\cdot r$ divides $|E_{\mathrm{tor}}(\mathbb{Q})|$. Then $a(p^{d-1})\neq
r^{v}$ unless $d=r$ for some $v\in\mathbb{N}$.

For odd weights $k\geq 3$ newforms, we have the following result.

######  Theorem 1.5.

Let $\gcd(n,2\cdot N)=1$. Then $a(p^{d-1})\neq\pm 1$ and for $n>1$, we also
have $a(n)\neq\pm 1$. Furthermore, if $a(n)=\pm\ell$ for some prime $\ell$,
then $n=p^{d-1}$ where $d|\ell(\ell^{2}-1)$ is odd. If $\pm\ell$ is not
defective, then $d$ is an odd prime.

In Section 3.2 and 4, we give results allowing us to state the above theorems
independently of the level.

##  2\. Preliminaries

###  2.1. Lucas Sequences and their primitive prime divisors

We recall the deep work of Bilu, Hanrot and Voutier [7] on Lucas sequences
which is central to this note.

A Lucas pair $(\alpha,\beta)$ is a pair of algebraic integers, roots of a
monic quadratic polynomial $F(x)=(x-\alpha)(x-\beta)\in\mathbb{Z}[x]$ where
$\alpha+\beta$, $\alpha\beta$ are coprime non-zero integers and such that
$\alpha/\beta$ is not a root of unity. To any Lucas pair $(\alpha,\beta)$ we
can associate a sequence of integers
$\\{u_{n}(\alpha,\beta)\\}=\\{u_{1}=1,u_{2}=\alpha+\beta,\dots\\}$ called
Lucas numbers defined by the following formula

(2.1) $u_{n}(\alpha,\beta):=\frac{\alpha^{n}-\beta^{n}}{\alpha-\beta}.$

We call a prime $\ell\mid u_{n}(\alpha,\beta)$ a primitive prime divisor of
$u_{n}(\alpha,\beta)$ if $\ell\nmid(\alpha-\beta)^{2}u_{1}(\alpha,\beta)\cdots
u_{n-1}(\alpha,\beta)$. We call a Lucas number $u_{n}(\alpha,\beta)$ with
$n>2$ defective111We do not consider the absence of a primitive prime divisor
for $u_{2}(\alpha,\beta)=\alpha+\beta$ to be a defect. if
$u_{n}(\alpha,\beta)$ does not have a primitive prime divisor. Bilu, Hanrot,
and Voutier [7] proved the following theorem for all Lucas sequences.

######  Theorem 2.1.

Every Lucas number $u_{n}(\alpha,\beta)$, with $n>30,$ has a primitive prime
divisor.

This theorem is sharp in the sense that there are sequences for which
$u_{30}(\alpha,\beta)$ does not have a primitive prime divisor. Their work,
combined with a subsequent paper222This paper included a few cases which were
omitted in [7]. of Abouzaid [1], gives the complete classification of
defective Lucas numbers in two categories; a sporadic family of examples and a
set of infinite parametrized families, as can be seen from Tables 1-4 in
Section 1 of [7] and Theorem 4.1 of [1]. The main arguments in our proofs will
largely rely on relative divisibility properties of Lucas numbers. We now
recall some of these facts333See Section 2 of [7]..

######  Proposition 2.2 (Prop. 2.1 (ii) of [7]).

If $d\mid n$, then $u_{d}(\alpha,\beta)|u_{n}(\alpha,\beta).$

In order to keep track of the first occurrence of a prime divisor, we define
$m_{\ell}(\alpha,\beta)$ to be the smallest $n\geq 2$ for which $\ell\mid
u_{n}(\alpha,\beta)$. We note that $m_{\ell}(\alpha,\beta)=2$ if and only if
$\alpha+\beta\equiv 0\pmod{\ell}.$

######  Proposition 2.3 (Cor. 2.2444This corollary is stated for Lehmer
numbers. The conclusions hold for Lucas numbers because
$\ell\nmid(\alpha+\beta)$. of [7]).

If $\ell\nmid\alpha\beta$ is an odd prime with $m_{\ell}(\alpha,\beta)>2$,
then the following are true.

  1. 1.

If $\ell\mid(\alpha-\beta)^{2}$, then $m_{\ell}(\alpha,\beta)=\ell.$

  2. 2.

If $\ell\nmid(\alpha-\beta)^{2}$, then $m_{\ell}(\alpha,\beta)\mid(\ell-1)$ or
$m_{\ell}(\alpha,\beta)\mid(\ell+1).$

######  Remark.

If $\ell|\alpha\beta$, then either $\ell|u_{n}(\alpha,\beta)$ for all $n$ or
$\ell\nmid u_{n}(\alpha,\beta)$ for all $n$.

We now recall the following facts about newforms of weight $k\in\mathbb{N}$
and character $\chi$ that we will denote by $S_{k}^{new}(\Gamma_{0}(N),\chi)$.
We suggest that the reader takes a look at the book of Cohen and Strömberg [9]
for a thorough introduction to the theory of modular forms and to the book of
Ono [16] for a clear and concise exposition to more advanced topics.

######  Proposition 2.4.

Suppose that $f(z)=q+\sum_{n\geq 2}a(n)q^{n}\in S_{k}(\Gamma_{0}(N),\chi)$ is
a normalized newform with nebentypus $\chi$. Then the following are true.

  1. 1.

If $\gcd(n_{1},n_{2})=1,$ then $a(n_{1}n_{2})=a(n_{1})a(n_{2}).$

  2. 2.

If $p\nmid N$ is prime and $m\geq 2$, then

$a(p^{m})=a(p)a(p^{m-1})-\chi(p)p^{k-1}a(p^{m-2}).$

  3. 3.

If $p\nmid N$ is prime and $\alpha_{p}$ and $\beta_{p}$ are roots of
$F_{p}(x):=x^{2}-a(p)x+\chi(p)p^{k-1},$ then

$a(p^{m})=u_{m+1}(\alpha_{p},\beta_{p})=\frac{\alpha_{p}^{m+1}-\beta_{p}^{m+1}}{\alpha_{p}-\beta_{p}}.$

Moreover, we have the Deligne’s bound $|a(p)|\leq 2p^{\frac{k-1}{2}}$.

In this note, we consider Lucas sequences arising from the roots of the
Frobenius polynomial

(2.2)
${}F_{p}(x):=x^{2}-Ax+B:=x^{2}-a(p)x+\chi(p)p^{k-1}=(x-\alpha_{p})(x-\beta_{p}),$

for a fixed prime $p\nmid N$ where

$u_{n}(\alpha_{p},\beta_{p}):=a(p^{n-1})=\frac{\alpha_{p}^{n}-\beta_{p}^{n}}{\alpha_{p}-\beta_{p}},$

and $|a(p)|\leq 2p^{\frac{k-1}{2}}$.

###  2.2. Modular Forms and their Galois Representation

######  Definition 2.5.

We say that a newform $f\in S_{k}^{new}(\Gamma_{0}(N),\chi)$ has trivial
residual mod 2 Galois representation if $a(p)$ is even for all $p\nmid 2\cdot
N$.

######  Remark.

The condition $p\neq 2$ comes from the fact that the determinant of the
representation of the Galois group evaluated at the Frobenius element needs to
be nonzero in order to be invertible. Using Proposition 2.4-(2), this implies
that we can derive $a(p^{d})$ to be odd if and only if $d$ is even. Similarly,
we get that $a(p^{d})$ is even if and only if $d$ is odd. It follows that
$a(n)$ is odd if and only if $n$ is an odd square. Furthermore, requiring
$f\in S_{2}^{new}(\Gamma_{0}(N))$ to have trivial residual mod $2$ Galois
representation is equivalent to asking that the associated modular elliptic
curve has a rational $2$-torsion point.

##  3\. Newforms of weight $k=2$

We begin by studying newforms of weight $k=2$, level $N$ and trivial character
$\chi$ with integer coefficients. These modular forms are interesting by
themselves as they correspond to modular elliptic curves. For completeness, we
recall some facts from the weight 2 case presented in [2].

######  Lemma 3.1 (Lemma 2.1 [2]).

Assume $a(p)$ is even for primes $p\nmid 2\cdot N$. The only defective odd
values $u_{d}(\alpha_{p},\beta_{p})$ are given in Table 1 by

$(d,A,B,k)\in\\{(3,2,3,2),(5,2,11,2)\\},$

and in rows 1 and 2 of Table 2.

######  Remark.

For $u_{3}(\alpha_{p},\beta_{p})=\varepsilon 3^{r}$, we have $3\nmid a(p)$ and
so $u_{3}(\alpha,\beta)$ is the first occurrence of $3$ in the sequence. If
$3|a(p)$ however, then $\varepsilon 3^{r}$ is no longer a defective value.

We will make use of the two following fundamental results.

######  Theorem 3.1 (Modularity Theorem [8]).

Let $E/\mathbb{Q}$ be an elliptic curve with conductor $N$ and a rational
$2$-torsion point. Denote the associated newform with trivial residual mod $2$
Galois representation by $f_{E}(z)=\sum_{n\geq 1}a(n)q^{n}\in
S_{2}^{New}(\Gamma_{0}(N))\cap\mathbb{Z}[[q]]$. Then for all primes $p\nmid
2\cdot N$ of good reduction, we have

$a(p)=p+1-\\#E(\mathbb{F}_{p}),$

where $\\#E(\mathbb{F}_{p})$ denotes the number of $\mathbb{F}_{p}$-points of
the elliptic curve reduced mod $p$.

The following theorem of Mazur classifies the possible torsion groups of
ellitpic curves of $E/\mathbb{Q}$.

######  Theorem 3.2 (Mazur’s Theorem [15]).

If $E/\mathbb{Q}$ is an elliptic curve, then

$E_{\mathrm{tor}}(\mathbb{Q})\in\\{\mathbb{Z}/N\mathbb{Z}\ |\ 1\leq N\leq 10\
\mathrm{or}\ N=12\\}\cup\\{\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}\
|\ N=2,4,6,8\\}.$

Furthermore, recall that if $E/\mathbb{Q}$ has good reduction at $p\nmid m$
for some $m\in\mathbb{N}$, then the reduction map

(3.1) $\pi\colon E(\mathbb{Q})\to E(\mathbb{F}_{p}),$

is injective when restricted to $m$-torsion [19]. As a consequence, we get the
following result.

######  Lemma 3.2 (Lemma 3.1 [2]).

Suppose that $E/\mathbb{Q}$ is an elliptic curve and that
$r\mid\\#E_{\mathrm{tor}}(\mathbb{Q})$. Then for all primes $p\nmid 2\cdot
r\cdot N$, we have

$a(p^{d})\equiv 1+p+p^{2}+\cdots+p^{d}\mod r.$

######  Lemma 3.3 (Lemma 3.2 [2]).

If $E/\mathbb{Q}$ has a rational $2$ and $r$-torsion point where $r=3,5$, then
for all $\gcd(n,2\cdot r\cdot N)=1$, we have $|a(n)|\neq 1$.

###  3.1. Integer points on Thue equations

We discuss the general approach to solve the equation $a(n)=\ell$ for some
prime $\ell$, positive or negative, where $a(n)$ is the Fourier coefficient of
$f\in S_{k}^{new}(\Gamma_{0}(N),\chi)\cap\mathbb{Z}[[q]]$. Assume that
$|a(n)|\neq 1$ for some given values of $n$. Using Proposition 2.4 (1)-(2), we
can see that studying $a(n)=\ell$ is equivalent to studying $a(p^{d-1})=\ell$
for $d\big{|}|\ell|(|\ell|-1)(|\ell|+1)$. From the two-term recurrence
relation satisfied by $a(p^{d-1})$, $a(p^{d-1})=\ell$ reduces to the search of
integer points on special curves. We make this statement precise now.

Let $D$ be a non-zero integer. A polynomial equation of the form $F(X,Y)=D$,
where $F(X,Y)\in\mathbb{Z}[X,Y]$ is a homogeneous polynomial, is called a Thue
equation. We will consider those equations arising from the series expansion
of

(3.2) $\frac{1}{1-\sqrt{Y}T+XT^{2}}=\sum_{m=0}^{\infty}F_{m}(X,Y)\cdot
T^{m}=1+\sqrt{Y}\cdot T+(Y-X)\cdot T^{2}+\dots$

######  Lemma 3.4.

If $a(n)$ satisfies Proposition 2.4, and $p\nmid N$ is a prime, then

$F_{2m}\left(\chi(p)p^{k-1},a(p)^{2}\right)=a(p^{2m}).$

Hence, solving $a(p^{2m})=q$ boils down to computing integer solutions to the
equation

$F_{2m}(X,Y)=\ell.$

Methods for solving Thue equations are implemented in Sage [5], Magma, and are
best suited for $m\geq 3$. For $m=1,2$, these equations often have infinitely
many solutions as they do not represent curves with positive genus when the
weight is $k=1,2$, hence we require extra information to infer their
finiteness. In the case of weight $2$ newforms, the idea is to introduce a $3$
and $5$-torsion point to get additional congruences to avoid having to deal
with infinitely many solutions for the equations
$a(p^{2})=\ell,\,a(p^{4})=\ell$. Indeed, note that

$d\big{|}|\ell|(|\ell|-1)(|\ell|+1)\equiv 0\ (\mathrm{mod}\ 3),$

for all $\ell$ and hence $d=3$ will always have to be checked.

###  3.2. Some Congruences

We now list congruences obtained using Lemma 3.2.

######  Lemma 3.5.

Let $E/\mathbb{Q}$ be an elliptic curve with conductor $N$ having a rational
$2$ and $3$-torsion point. Consider primes $p$ for which $\gcd(p,2\cdot 3\cdot
N)=1$.

  * 1.

If $a(p^{d-1})=\ell=\pm 3$, then

$(p,d)\in\\{(1,0),(2,0),(2,2)\ (\mathrm{mod}\ 3)\\}.$

  * 2.

If $a(p^{d-1})=\ell\equiv 1\ (\mathrm{mod}\ 3)$, then

$(p,d)\in\\{(1,1),(2,\text{odd})\ (\mathrm{mod}\ 3)\\}.$

  * 3.

If $a(p^{d-1})=\ell\equiv 2[3]$, then

$(p,d)=(1,2)\ (\mathrm{mod}\ 3).$

######  Remark.

In point 2, the last pair is problematic as $d$ is always odd. Hence, it is
not possible to provide a general result in this case.

Let $E/\mathbb{Q}$ be an elliptic curve with conductor $N$ having a rational
$2$ and $5$-torsion point. Consider primes $p$ for which $\gcd(p,2\cdot 5\cdot
N)=1$.

  * 1.

If $a(p^{d-1})=\pm 5$, then

$(p,d)\in\\{(1,0),(3,0),(3,4),(4,0),(4,4),(4,2)\ (\mathrm{mod}\ 5)\\}.$

  * 2.

If $a(p^{d-1})=\ell\equiv 1\ (\mathrm{mod}\ 5)$, then

$(p,d)\in\\{(1,1),(2,1),(3,1),(4,1),(4,3)\ (\mathrm{mod}\ 5)\\}.$

  * 3.

If $a(p^{d-1})=\ell\equiv 2\ (\mathrm{mod}\ 5)$, then

$(p,d)\in\\{(1,2),(2,3)\ (\mathrm{mod}\ 5)\\}.$

  * 4.

If $a(p^{d-1})=\ell\equiv 3\ (\mathrm{mod}\ 5)$, then

$(p,d)\in\\{(1,3),(3,3),(2,2)\ (\mathrm{mod}\ 5)\\}.$

  * 5.

If $a(p^{d-1})=\ell\equiv 4\ (\mathrm{mod}\ 5)$, then

$(p,d)\in\\{(1,4),(3,2)\ (\mathrm{mod}\ 5)\\}.$

Using the above congruences extensively as well as the relative divisibility
property of Lucas numbers, we get the following result.

######  Theorem 3.3.

Suppose $f(z)=q+\sum_{n\geq 2}a(n)q^{n}\in
S_{2}^{new}(\Gamma_{0}(N))\cap\mathbb{Z}[[q]]$ has trivial residual mod 2
Galois representation, namely, $E/\mathbb{Q}$ is an elliptic curve of
conductor $N$ with a rational $2$-torsion point. Then the following are true.

  1. If $E/\mathbb{Q}$ has a rational $3$-torsion point, then for $n>1$ and $\gcd(n,2\cdot 3\cdot N)=1$, we have

     * 1.

If $a(n)=7,13,19,31,37$, then $n=p^{2}$ with $p=2\ (\mathrm{mod}\ 3)$.

     * 2.

If $a(n)=29$ then $n=p^{d-1}=13^{4}$ and $a(p)=\pm 2$.

     * 3.

If $a(n)=41$ then $n=p^{d-1}=43^{4}$ and $a(p)=\pm 4$.

     * 4.

If $a(n)=-19$ then $n=p^{d-1}=7^{4}$ and $a(p)=\pm 2$.

     * 5.

If $a(n)=-31$ then $n=p^{d-1}=7^{4}$ and $a(p)=\pm 4$.

     * 6.

If $a(n)=-79$ then $n=p^{d-1}=167^{4}$ and $a(p)=\pm 8$.

Furthermore,

$a(n)\not\in\\{-1,1,5,-7,11,-13,17,23,-37,-43,47,53,59,-61,-67,71,-73,83,89,-97\\}.$

  2. If $E/\mathbb{Q}$ has a rational $5$-torsion point, then for $n>1$ and $\gcd(n,2\cdot 5\cdot N)=1$, we have

     * 1.

If $\ell\equiv 1\ (\mathrm{mod}\ 5)$ and $a(n)=\ell$, then $n=p^{2}$ and
$p\equiv 4\ (\mathrm{mod}\ 5)$.

     * 2.

If $\ell\equiv 2\ (\mathrm{mod}\ 5)$, $\ell\neq-3$ and $a(n)=\ell$, then
$n=p^{2}$ and $p\equiv 2\ (\mathrm{mod}\ 5)$.

     * 3.

If $\ell\equiv 3\ (\mathrm{mod}\ 5)$, $\ell\neq 3$ and $a(n)=\ell$, then
$n=p^{2}$ and $p\equiv 1,3\ (\mathrm{mod}\ 5)$.

Furthermore,

$a(n)\not\in\\{-1,1,-11,19,29,-31,-41,59,-61,-71,79,89,-691\\}.$

######  Remark.

In the first part of the theorem, we have omitted the primes
$43,61,67,73,79,97$, which are of the form $\ell\equiv 1\ (\mathrm{mod}\ 3)$,
and the primes $-\ell\equiv 1\ (\mathrm{mod}\ 3)$ due to the large number of
curves involved. However, following the methods outlined in this text, the
interested reader will have no difficulty investigating these cases. In the
second part, note that the primes of the form $\ell\equiv 4\ (\mathrm{mod}\
5)$ are those of the outlined list. We’ve also ruled out $-691$ simply because
it is a pretty number and a nice example of application of Theorem 3.5.

######  Theorem 3.4.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ with a $2$ and
$3$-torsion point. Let $n>1$ and $gcd(n,2\cdot 3\cdot N)=1$. If $\ell\equiv 2\
(\mathrm{mod}\ 3),\ell\neq 5$ and the odd prime divisors $d$ of
$|\ell|(|\ell|-1)(|\ell|+1)$ are not congruent to $2\ (\mathrm{mod}\ 3)$, then
$a(n)\neq\ell$.

######  Theorem 3.5.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ with a $2$ and
$5$-torsion point. Let $n>1$ and $\gcd(n,2\cdot 5\cdot N)=1$.

  * 1.

If $\ell\equiv 1\ (\mathrm{mod}\ 5)$ and the odd prime divisors $d$ of
$|\ell|(|\ell|-1)(|\ell|+1)$ are not congruent to $1,3\ (\mathrm{mod}\ 5)$,
then $a(n)\neq\ell$.

  * 2.

If $\ell\equiv 2\ (\mathrm{mod}\ 5),\ell\neq-3$ and the odd prime divisors $d$
of $|\ell|(|\ell|-1)(|\ell|+1)$ are not congruent to $2,3\ (\mathrm{mod}\ 5)$,
then $a(n)\neq\ell$.

  * 3.

If $\ell\equiv 3\ (\mathrm{mod}\ 5),\ell\neq 3$ and the odd prime divisors $d$
of $|\ell|(|\ell|-1)(|\ell|+1)$ are not congruent to $2,3\ (\mathrm{mod}\ 5)$,
then $a(n)\neq\ell$.

  * 4.

If $\ell\equiv 4\ (\mathrm{mod}\ 5)$ and the odd prime divisors $d$ of
$|\ell|(|\ell|-1)(|\ell|+1)$ are not congruent to $2,4\ (\mathrm{mod}\ 5)$,
then $a(n)\neq\ell$.

The above theorems can be made independent of the level using the following
lemma.

######  Lemma 3.6.

Let $p|N$ be a prime and $N$ the level of the newform $f(z)$. Then

$a_{f}(p^{m})=a(p)a(p^{m-1})=\begin{cases}(\pm 1)^{m}\ \ \ \ \ &{\text{\rm
if}}\ \mathrm{ord}_{p}(N)=1,\\\ 0\ \ \ \ \ &{\text{\rm if}}\
\mathrm{ord}_{p}(N)\geq 2.\end{cases}$

######  Theorem 3.6.

Let $E/\mathbb{Q}$ be an elliptic curve with conductor $N$ and $f(z)$ the
corresponding newform with Fourier coefficients $a(n)$. For $r=3,5,$ suppose
that $2\cdot r$ divides $|E_{\mathrm{tor}}(\mathbb{Q})|$. Then $a(p^{d-1})\neq
r^{v}$ unless $d=r$ for some $v\in\mathbb{N}$.

###### Proof.

Let $r=3$. Then by Lemma 3.5, $3\big{|}|a(p^{d-1})|$ if and only if $3|d$.
Indeed, if $p\equiv 0,2\ (\mathrm{mod}\ 3)$, then $a(p^{d-1})\equiv 1\
(\mathrm{mod}\ 3)$ and so $p\equiv 1\ (\mathrm{mod}\ 3)$ implies that $3|d$.
Suppose that $a(p^{d-1})=3^{v}$ and $d>3$, then $a(p^{d-4})$ is also a
multiple of $3$, which contradicts that $3^{v}$ is not defective. Thus $d=3$
is the only solution.

Let $r=5$. Then by Lemma 3.5, $5\big{|}|a(p^{d-1})|$ if and only if $p\equiv
1[5]$ and $5|d$. For $d>5$, we have $a(p^{d-6})$ is also a multiple of $5$,
violating that $a(p^{d-1})$ is not defective. Hence $d=5$. ∎

##  4\. Newforms of odd weight $k\geq 3$

In this section, we explain the basic framework to rule out odd prime values
$\pm\ell$ as Fourier coefficients of odd weight $k\geq 3$ normalized newforms
with integer coefficients of level $N$ and nebentypus $\chi$ given by a
quadratic Dirichlet character and trivial residual mod $2$ Galois
representation.

######  Theorem 4.1.

Let $\gcd(n,2\cdot N)=1$. Then $a(p^{d-1})\neq\pm 1$ and for $n>1$ we have
$a(n)\neq\pm 1$. It follows that if $a(n)=\pm\ell$ for some prime $\ell$, then
$n=p^{d-1}$ where $d|\ell(\ell^{2}-1)$ is odd. If $\pm\ell$ is not defective,
then $d$ is an odd prime.

###### Proof.

5Note that $\pm 1$ is a defective value which must be located in the first 2
rows of Table 4. In fact, the table of sporadic values does not have to be
considered. In row 1 of Table 4, we have that if $a(p^{d-1})=-1$, then it must
be at $u_{3}=a(p^{2})=-1$, where the constraints imply that we must satisfy
$\chi(p)p^{k-1}=a(p)^{2}+1$. Since $k-1$ is even, let’s write $k-1=2m$. Then
we’re left with the equation $\chi(p)x^{2m}=y^{2}+1$ and clearly there are no
solutions for $\chi(p)=0,-1$. If $\chi(p)=1$ then we obtain
$(x^{m})^{2}-y^{2}=1$ and this gives us the integer solutions $x=\pm 1,y=0$
which aren’t allowed. Hence $a(p^{d-1})\neq-1$ for $\gcd(p,2\cdot N)=1$. Now
for row 2, if $a(p^{d-1})=1$ then it must happen for $u_{3}=a(p^{2})=1$ with
constraints given by $\chi(p)p^{k-1}=a(p)^{2}-1$,$a(p)>1$. There are no
solutions to this equation for $\chi(p)=0,-1,1$. Hence $a(p^{d-1})\neq\pm 1$.
∎

######  Lemma 4.1.

The curve $a(p^{2})=\pm\ell$ has no solutions if $\chi(p)=0$. If $\chi(p)=1$
then the curve has the form $(y-x^{m})(y+x^{m})=\pm\ell$ which has a unique
solution depending on $\ell$ only. For $\chi(p)=-1$ the curve has the form
$y^{2}+(x^{m})^{2}=\pm\ell$ which has no solutions for $-\ell$ and has
finitely many solutions for $+\ell$ as it is a sum of squares.

Hence, to rule out or locate any odd prime value $\ell$ as a Fourier
coefficient of $f(z)$, it suffices to follow the following steps. Let
$\gcd(n,2\cdot N)=1$ and $\ell$ be an odd prime.

  * 1.

By multiplicativity of the Fourier coefficients, we have that

$a(n)=\pm\ell\text{ if and only if }\prod_{i}a(p_{i}^{d_{i}-1})=\pm\ell.$

  * 2.

Use Theorem 4.1.

  * 3.

Use the Sage Thue solver [5] to solve $a(p^{d-1})=\pm\ell$ and analyze the
solutions.

######  Lemma 4.2 (Proposition 13.3.14 [9]).

Let $p|N$ and assume that $\chi$ can be defined modulo $N/p$ and let $f=\sum
a(n)q^{n}\in S_{k}^{new}(\Gamma_{0}(N),\chi)$ be a normalized eigenform. Then

  * 1.

If $p^{2}|N$, then $a(p)=0$.

  * 2.

If $p^{2}\nmid N$, then $a(p)^{2}=\chi_{1}(p)p^{k-2}$ where $\chi_{1}$ is the
character modulo $N/p$ equivalent to $\chi$.

Using this lemma, we easily obtain the following.

######  Corollary 4.2.

Suppose that $\chi$ can be reduced modulo $N/p$ and call it $\chi_{1}$. Then
$a(p^{m})=0$ for all $m\geq 1$ and $a(n)=0$ for all $(n,N)\geq p$.

###### Proof.

Recall that $a(p^{m})=a(p)a(p^{m-1})$ and if $p^{2}|N$ the result follows
immediately. For $p^{2}\nmid N$ we have $a(p)^{2}=\chi_{1}(p)p^{k-2}$. Since
$k$ is odd, $k-2$ is odd and if $\chi_{1}(p)\neq 0$ then $a(p)$ cannot be an
integer which is what we require for our newforms. Hence $a(p)=0$ is the only
possibility. ∎

######  Lemma 4.3 (Corollary 13.3.17 [9]).

Let $p|N$ and assume that $\chi$ cannot be defined modulo $N/p$. If $f\in
S_{k}^{new}(\Gamma_{0}(N),\chi)$ is a normalized eigenform, then
$|a(p)|=p^{\frac{k-1}{2}}$.

######  Corollary 4.3.

Let $p|N$ and assume that $\chi$ cannot be defined modulo $N/p$. Then
$a(p^{m})=(\pm 1)^{m}(p^{\frac{k-1}{2}})^{m}$.

##  5\. Newforms of weight $k=1$

In this section, we discuss briefly the problems arising in the case of weight
$k=1$ newforms. Recall that Proposition 2.2 is the main tool to determine if
$d$ is prime or not. However, we may have that $n|d$ and $u_{n}=-1$. From the
first row of the defective table, we know that this may happen only if $n=3$,
namely if $u_{3}=a(p^{2})=-1$. Thus, we would like to avoid $3|d$ but that is
not possible as $\ell(\ell-1)(\ell+1)$ is always divisible by 3. This implies
that we only have the information that $d|\ell(\ell-1)(\ell+1)$ is odd and no
longer an odd prime. First, this leads us to a large amount of equations to
verify in order to rule out an odd prime as a Fourier coefficient. Secondly,
when $d=3$, the equation that we obtain is

$y^{2}=\pm\ell+\chi(p)$

and is incredibly difficult to solve as it is still an open problem to
determine if there are infinitely many primes of the form $\ell=y^{2}+1$.

##  6\. Appendix

$(A,B)$ | Defective $u_{n}(\alpha,\beta)$
---|---
$(\pm 1,2^{1})$ |  $u_{5}=-1$, $u_{7}=7$, $u_{8}=\mp 3$, $u_{12}=\pm 45$,
$u_{13}=-1$, $u_{18}=\pm 85$, $u_{30}=\mp 24475$
$(\pm 1,3^{1})$ |  $u_{5}=1$, $u_{12}=\pm 160$
$(\pm 1,5^{1})$ |  $u_{7}=1$, $u_{12}=\mp 3024$
$(\pm 2,3^{1})$ |  $u_{3}=1$, $u_{10}=\mp 22$
$(\pm 2,7^{1})$ | $u_{8}=\mp 40$
$(\pm 2,11^{1})$ | $u_{5}=5$
$(\pm 4,5^{1})$ | $u_{6}=\pm 44$
$(\pm 5,7^{1})$ | $u_{10}=\mp 3725$
$(\pm 3,2^{3})$ | $u_{3}=1$
$(\pm 5,2^{3})$ | $u_{6}=\pm 85$
Table 1. Sporadic family of defective $u_{n}(\alpha,\beta)$ satisfying equation 2.2 in even weight $2k$ including $2k=2$ [4]. $(A,B)$ | Defective $u_{n}(\alpha,\beta)$
---|---
$(\pm 1,-1)$ | $u_{5}=5,u_{12}=\pm 144$
$(\pm 1,2^{1})$ |  $u_{5}=-1$, $u_{7}=7$, $u_{8}=\mp 3$, $u_{12}=\pm 45$,
$u_{13}=-1$, $u_{18}=\pm 85$, $u_{30}=\mp 24475$
$(\pm 1,2^{2})$ |  $u_{5}=5$,$u_{12}=\mp 231$
$(\pm 1,3^{1})$ |  $u_{5}=1$, $u_{12}=\pm 160$
$(\pm 1,5^{1})$ |  $u_{7}=1$, $u_{12}=\mp 3024$
$(\pm 2,3^{1})$ | $u_{10}=\mp 22$
$(\pm 2,7^{1})$ | $u_{8}=\mp 40$
$(\pm 2,11^{1})$ | $u_{5}=5$
$(\pm 5,7^{1})$ | $u_{10}=\mp 3725$
Table 2. Sporadic family of defective $u_{n}(\alpha,\beta)$ satisfying equation 2.2 in odd weight $k\geq 1$. $(A,B)$ |  Defective $u_{n}(\alpha,\beta)$ | Constraints on parameters
---|---|---
$(\pm m,p)$ | $u_{3}=-1$ |  $m>1$ and $p=m^{2}+1$
$(\pm m,p^{2k-1})$ | $u_{3}=\varepsilon 3^{r}$ | $\begin{aligned} &\ \ \ \ \ \ (p,\pm m)\in B_{1,k}^{r,\varepsilon}\text{ with }3\nmid m,\\\ &(\varepsilon,r,m)\neq(1,1,2),\text{ and }m^{2}\geq 4\varepsilon 3^{r-1}\end{aligned}$
$(\pm m,p^{2k-1})$ | $u_{4}=\mp m$ |  $(p,\pm m)\in B_{2,k}$ with $m>1$ odd
$(\pm m,p^{2k-1})$ | $u_{4}=\pm 2{\color[rgb]{0,0,0}\varepsilon}m$ | $\begin{aligned} (p,\pm m)\in B_{3,k}^{\varepsilon}&\text{ with }{\color[rgb]{0,0,0}(\varepsilon,m)\not=(1,2)}\\\ &\text{ and }m>2\text{ even}\end{aligned}$
$(\pm m,p^{2k-1})$ | $u_{6}={\color[rgb]{0,0,0}\pm(-2)^{r}m(2m^{2}+(-2)^{r})/3}$ | $\begin{aligned} (p,&\pm m)\in B_{4,k}^{r}\text{ with }\gcd(m,6)=1,\\\ &{\color[rgb]{0,0,0}(r,m)\not=(1,1)},\text{ and }{\color[rgb]{0,0,0}m^{2}\geq(-2)^{r+2}}\end{aligned}$
$(\pm m,p^{2k-1})$ | ${\color[rgb]{0,0,0}u_{6}=\pm\varepsilon m(2m^{2}+3\varepsilon)}$ |  $(p,\pm m)\in B_{5,k}^{\varepsilon}$ with $3\mid m$ and $m>3$
$(\pm m,p^{2k-1})$ | $u_{6}=\pm 2^{r+1}{\color[rgb]{0,0,0}\varepsilon}m(m^{2}+3{\color[rgb]{0,0,0}\varepsilon}\cdot 2^{r-1})$ | $\begin{aligned} (p,\pm m)\in B_{6,k}^{r,\varepsilon}\text{ with }m\equiv 3\bmod{6}\\\ \text{and }m^{2}\geq 3{\color[rgb]{0,0,0}\varepsilon}\cdot 2^{r+2}\end{aligned}$
Table 3. Parameterized family of defective $u_{n}(\alpha,\beta)$ satisfying
equation 2.2 in even weight $2k\geq 2$ [4].
Notation: $m,k,r\in\mathbb{Z}^{+}$, $\varepsilon=\pm 1$, $p$ is a prime number. $(A,B)$ |  Defective $u_{n}(\alpha,\beta)$ | Constraints on parameters
---|---|---
$(\pm m,\chi(p)p^{k-1})$ | $u_{3}=-1$ | $\chi(p)p^{k-1}=m^{2}+1$
$(\pm m,\chi(p)p^{k-1})$ | $u_{3}=1$ | $\chi(p)p^{k-1}=m^{2}-1\text{ with }m>1$
$(\pm m,\chi(p)p^{k-1})$ | $u_{3}=\varepsilon 3^{r}$ | $\begin{aligned} &\ \ \ \ \ \ \chi(p)p^{k-1}=m^{2}-\varepsilon 3^{r}\text{ with }3\nmid m,\\\ &(\varepsilon,r,m)\neq(1,1,2),\text{ and }r>0\end{aligned}$
$(\pm m,\chi(p)p^{k-1})$ | $u_{4}=\pm\varepsilon m$ | $2\chi(p)p^{k-1}=m^{2}-\varepsilon\text{ with }2\nmid m,m\neq 1$
$(\pm m,\chi(p)p^{k-1})$ | $u_{4}=\pm 2\varepsilon m$ | $2\chi(p)p^{k-1}=m^{2}-2\varepsilon\text{ with }2|m,(\varepsilon,m)\neq(1,2)$
$(\pm m,\chi(p)p^{k-1})$ | $u_{6}=(\pm 2m^{3}\pm m)/3$ | $\begin{aligned} &\ \ \ \ \ \ 3\chi(p)p^{k-1}=m^{2}-1\text{ with }3\nmid m>3\end{aligned}$
$(\pm m,\chi(p)p^{k-1})$ | $u_{6}=\pm 2\varepsilon m^{3}\pm 3m$ | $3\chi(p)p^{k-1}=m^{2}-3\varepsilon\text{ with }3\mid m$
$(\pm m,\chi(p)p^{k-1})$ | $u_{6}=(\pm 2m^{3}(-2)^{r}\pm m((-2)^{r})^{2})/3$ | $\begin{aligned} &\ \ \ \ \ \ 12\chi(p)p^{k-1}=4m^{2}-(-2)^{r+2}\\\ &r>0,m\equiv\pm 1[6],(r,m)\neq(1,1)\end{aligned}$
$(\pm m,\chi(p)p^{k-1})$ | $u_{6}=\pm 2m^{3}\varepsilon\cdot 2^{r}\pm 3m(2^{r})^{2}$ | $12\chi(p)p^{k-1}=4m^{2}-3\cdot 2^{r+2}\varepsilon,r>0,m\equiv 3[6]$
Table 4. Parameterized family of defective $u_{n}(\alpha,\beta)$ satisfying
equation 2.2 in odd weight.
Notation: $m,k,r\in\mathbb{Z}^{+}$, $\varepsilon=\pm 1$, $p$ is a prime
number.

## References

  * [1] M. Abouzaid, _Les nombres de Lucas et Lehmer sans diviseur primitif_ , J. Théor. Nombres Bordeaux 18, 299-313 (2006).
  * [2] Malik Amir and Letong Hong, _On L-functions of modular elliptic curves and certain K3 surfaces_ , https://doi.org/10.1007/s11139-021-00388-w, Ramanujan Journal (2021).
  * [3] J. S. Balakrishnan, W. Craig, and K. Ono, _Variations of Lehmer’s Conjecture for Ramanujan’s tau-function_ , ISSN 0022-314X, Journal of Number Theory (2020).
  * [4] J. S. Balakrishnan, W. Craig, K. Ono, and W.-L. Tsai, _Variants of Lehmer’s speculation for newforms_ , arXiv:2005.10354 [math.NT] (2020).
  * [5] J. S. Balakrishnan, W. Craig, and K. Ono, _Sage code_ , https://github.com/jbalakrishnan/Lehmer.
  * [6] Y. Bilu and G. Hanrot, _Solving the Thue equations of high degree_ , Journal of Number Theory, 60, 373-392 (1996).
  * [7] Y. Bilu, G. Hanrot, and P. M. Voutier, _Existence of primitive divisors of Lucas and Lehmer numbers_ , J. Reine Angew. Math. 539, 75-122 (2001).
  * [8] C. Breuil, B. Conrad, F. Diamond, R. Taylor, _On the modularity of elliptic curves over Q: wild 3-adic exercises_ , Journal of the American Mathematical Society 14 (4): 843–939 (2001).
  * [9] H. Cohen and F. Strömberg, _Modular Forms, A Classical Approach_ , American Mathematical Society, Graduate Studies in Mathematics 179 (2017).
  * [10] S. Dembner and V. Jain, _Hyperelliptic curves and newform coefficients_ , arXiv:2007.08358 [math.NT] (2020), submitted for publication.
  * [11] M. Hanada and R. Madhukara, _Fourier coefficients of level 1 Hecke eigenforms_ , arXiv:2007.08683 [math.NT] (2020), submitted for publication.
  * [12] D. H. Lehmer, _The vanishing of Ramanujan’s $\tau(n)$_, Duke Math. J. 14, 429-433 (1947).
  * [13] The LMFDB Collaboration, The $L$-functions and Modular Forms Database, _http://lmfdb.org/_ , 2020,[Online, accessed July 2020].
  * [14] V. K. Murty, R. Murty, and N. Shorey, _Odd values of the Ramanujan tau function_ , Bull. Soc. Math. France 115 (1987), 391-395.
  * [15] B. Mazur _Rational isogenies of prime degree (with an appendix by D. Goldfeld)_ , Invent. Math., 44(2) : 129-162 (1978).
  * [16] K. Ono, _The Web of Modularity : Arithmetic of the Coefficients of Modular Forms and $q$-series_, American Mathematical Society, CBMS regional conference series in mathematics, ISSN 0160-7642; no. 102 (2003).
  * [17] S. Ramanujan, _On certain arithmetical functions_ , Trans. Camb. Phil. Soc., 22, 159–184 (1916).
  * [18] K. Ribet, _Galois representations attached to eigenforms with nebentypus_ , Lecture Notes in Math., vol. 601, Springer-Verlag, New York, pp.17-51 (1976).
  * [19] J. H. Silverman, _The Arithmetic of Elliptic Curves_ , Second Edition, New York: SpringerVerlag (2009).