A short note on inadmissible coefficients of weight 2 and 2k+1 newforms
aa r X i v : . [ m a t h . N T ] F e b A SHORT NOTE ON INADMISSIBLE COEFFICIENTS OF WEIGHT AND k + 1 NEWFORMS
MALIK AMIR AND ANDREAS HATZIILIOU
Abstract.
Let f ( z ) = q + P n ≥ a ( n ) q n be a weight k normalized newform with integercoefficients and trivial residual mod 2 Galois representation. We extend the results of Amirand Hong in [2] for k = 2 by ruling out or locating all odd prime values | ℓ | <
100 of theirFourier coefficients a ( n ) when n satisfies some congruences. We also study the case of oddweights k ≥ Introduction and Statement of the Results
In an article entitled “On certain arithmetical functions” , Ramanujan introduced the τ -function in 1916, known as the Fourier coefficients of the weight 12 modular form∆( z ) = q ∞ Y n =1 (1 − q n ) := ∞ X n =1 τ ( n ) q n = q − q + 252 q − q + 4830 q − ... where throughout q := e πiz . It was conjectured by Ramanujan that the τ -function is multi-plicative and this offered a glimpse into a much more general theory known today as the theoryof Hecke operators. Despite its importance in the large web of mathematics and physics, basicproperties of τ ( n ) are still unknown. The most famous example is Lehmer’s conjecture aboutthe nonvanishing of τ ( n ). Lehmer proved that if τ ( n ) = 0, then n must be a prime [12]. Onemay be interested in studying odd values taken by τ ( n ) or, more generally, the coefficients ofany newform. This is the question we consider as a variation of Lehmer’s original speculation.For an odd number α , Murty, Murty and Shorey [14] proved using linear forms in logarithmsthat τ ( n ) = α for all n sufficiently large. However, the bounds that they obtained are hugeand 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 τ ( n ) = ℓ has no solution for | ℓ | <
100 an odd prime. In addition, Hanada and Madhukaraproved in [11] that τ ( n ) = α has no solution for | α | <
100 an odd integer. Following theseideas, Amir and Hong investigated weight 2 and 3 newforms corresponding to modular ellipticcurves and a special family of K L -functions ofmodular elliptic curves and give a procedure to rule out odd prime values ℓ , positive or negative,as coefficients of any normalized newform of odd weight k ≥ Key words and phrases.
Lucas Sequences, Lehmer’s Conjecture, Modular Forms, L -Functions, EllipticCurves. rest of this paper, whenever we say newform of weight k , we talk about a newform with theaforementioned properties. In the case of weight 2 newforms, we have the following results. Theorem 1.1.
Suppose f ( z ) = q + P n ≥ a ( n ) q n ∈ S new (Γ ( N )) ∩ Z [[ q ]] has trivial residualmod 2 Galois representation, namely, E/ Q is an elliptic curve of conductor N with a rational -torsion point. Then the following are true.(1) If E/ Q has a rational -torsion point, then for n > and gcd( n, · · N ) = 1 , we havea. If a ( n ) = 7 , , , , , then n = p and p = 2[3] .b. If a ( n ) = 29 then n = p d − = 13 and a ( p ) = ± .c. If a ( n ) = 41 then n = p d − = 43 and a ( p ) = ± .d. If a ( n ) = − then n = p d − = 7 and a ( p ) = ± .e. If a ( n ) = − then n = p d − = 7 and a ( p ) = ± .f. If a ( n ) = − then n = p d − = 167 and a ( p ) = ± .Furthermore, a ( n )
6∈ {− , , , − , , − , , , − , − , , , , − , − , , − , , , − } . (2) If E/ Q has a rational -torsion point, then for n > and gcd( n, · · N ) = 1 , we havea. If ℓ ≡ and a ( n ) = ℓ , then n = p and p ≡ .b. If ℓ ≡ , ℓ = − and a ( n ) = ℓ , then n = p and p ≡ .c. If ℓ ≡ , ℓ = 3 and a ( n ) = ℓ , then n = p and p ≡ , .Furthermore, a ( n )
6∈ {− , , − , , , − , − , , − , − , , , − } . Theorem 1.2.
Let E/ Q be an elliptic curve of conductor N with a and -torsion point. Let n > and gcd ( n, · · N ) = 1 . If ℓ ≡ , ℓ = 5 and the odd prime divisors d of | ℓ | ( | ℓ |− | ℓ | +1) are not congruent to , then a ( n ) = ℓ . Theorem 1.3.
Let E/ Q be an elliptic curve of conductor N with a and -torsion point. Let n > and gcd( n, · · N ) = 1 .1. If ℓ ≡ and the odd prime divisors d of | ℓ | ( | ℓ | − | ℓ | + 1) are not congruent to , ,then a ( n ) = ℓ .2. If ℓ ≡ , ℓ = − and the odd prime divisors d of | ℓ | ( | ℓ | − | ℓ | + 1) are not congruentto , , then a ( n ) = ℓ .3. If ℓ ≡ , ℓ = 3 and the odd prime divisors d of | ℓ | ( | ℓ | − | ℓ | + 1) are not congruentto , , then a ( n ) = ℓ .4. If ℓ ≡ and the odd prime divisors d of | ℓ | ( | ℓ | − | ℓ | + 1) are not congruent to , ,then a ( n ) = ℓ . Theorem 1.4.
Let E/ Q be an elliptic curve with conductor N and f the corresponding newformwith Fourier coefficients a ( n ) . For r = 3 , , suppose that · r divides | E tor ( Q ) | . Then a ( p d − ) = r v unless d = r . For odd weights k ≥ Theorem 1.5.
Let gcd( n, · N ) = 1 . Then a ( p d − ) = ± and for n > , we also have a ( n ) = ± . Furthermore, if a ( n ) = ± ℓ for some prime ℓ , then n = p d − where d | ℓ ( ℓ − isodd. If ± ℓ is not defective, then d is an odd prime. SHORT NOTE ON INADMISSIBLE COEFFICIENTS OF WEIGHT 2 AND 2 k + 1 NEWFORMS 3 In Section 3.2 and 4, we give results allowing us to state the above theorems independentlyof the level. 2.
Preliminaries
Lucas Sequences and their primitive prime divisors.
We recall the deep work ofBilu, Hanrot and Voutier [7] on Lucas sequences which is central to this note.A
Lucas pair ( α, β ) is a pair of algebraic integers, roots of a monic quadratic polynomial F ( x ) = ( x − α )( x − β ) ∈ Z [ x ] where α + β , αβ are coprime non-zero integers and such that α/β is not a root of unity. To any Lucas pair ( α, β ) we can associate a sequence of integers { u n ( α, β ) } = { u = 1 , u = α + β, . . . } called Lucas numbers defined by the following formula(2.1) u n ( α, β ) := α n − β n α − β . We call a prime ℓ | u n ( α, β ) a primitive prime divisor of u n ( α, β ) if ℓ ∤ ( α − β ) u ( α, β ) · · · u n − ( α, β ).We call a Lucas number u n ( α, β ) with n > defective if u n ( α, β ) does not have a primitiveprime divisor. Bilu, Hanrot, and Voutier [7] proved the following theorem for all Lucas se-quences. Theorem 2.1.
Every Lucas number u n ( α, β ) , with n > , has a primitive prime divisor. This theorem is sharp in the sense that there are sequences for which u ( α, β ) does nothave a primitive prime divisor. Their work, combined with a subsequent paper of Abouzaid[1], gives the complete classification of defective Lucas numbers in two categories; a sporadicfamily of examples and a set of infinite parametrized families, as can be seen from Tables 1-4in Section 1 of [7] and Theorem 4.1 of [1]. The main arguments in our proofs will largely relyon relative divisibility properties of Lucas numbers. We now recall some of these facts . Proposition 2.2 (Prop. 2.1 (ii) of [7]) . If d | n , then u d ( α, β ) | u n ( α, β ) . In order to keep track of the first occurrence of a prime divisor, we define m ℓ ( α, β ) to be thesmallest n ≥ ℓ | u n ( α, β ). We note that m ℓ ( α, β ) = 2 if and only if α + β ≡ ℓ ) . Proposition 2.3 (Cor. 2.2 of [7]) . If ℓ ∤ αβ is an odd prime with m ℓ ( α, β ) > , then thefollowing are true.(1) If ℓ | ( α − β ) , then m ℓ ( α, β ) = ℓ. (2) If ℓ ∤ ( α − β ) , then m ℓ ( α, β ) | ( ℓ − or m ℓ ( α, β ) | ( ℓ + 1) . Remark. If ℓ | αβ , then either ℓ | u n ( α, β ) for all n or ℓ ∤ u n ( α, β ) for all n . We now recall the following facts about newforms of weight k ∈ N and character χ that wewill denote by S newk (Γ ( N ) , χ ). We suggest that the reader takes a look at the book of Cohenand Str¨omberg [9] for a thorough introduction to the theory of modular forms and to the bookof Ono [16] for a clear and concise exposition to more advanced topics. We do not consider the absence of a primitive prime divisor for u ( α, β ) = α + β to be a defect. This paper included a few cases which were omitted in [7]. See Section 2 of [7]. This corollary is stated for Lehmer numbers. The conclusions hold for Lucas numbers because ℓ ∤ ( α + β ). MALIK AMIR AND ANDREAS HATZIILIOU
Proposition 2.4.
Suppose that f ( z ) = q + P n ≥ a ( n ) q n ∈ S k (Γ ( N ) , χ ) is a normalized new-form with nebentypus χ . Then the following are true.(1) If gcd( n , n ) = 1 , then a ( n n ) = a ( n ) a ( n ) . (2) If p ∤ N is prime and m ≥ , then a ( p m ) = a ( p ) a ( p m − ) − χ ( p ) p k − a ( p m − ) . (3) If p ∤ N is prime and α p and β p are roots of F p ( x ) := x − a ( p ) x + χ ( p ) p k − , then a ( p m ) = u m +1 ( α p , β p ) = α m +1 p − β m +1 p α p − β p . Moreover, we have the Deligne’s bound | a ( p ) | ≤ p k − . In this note, we consider Lucas sequences arising from the roots of the Frobenius polynomial(2.2) F p ( x ) := x − Ax + B := x − a ( p ) x + χ ( p ) p k − = ( x − α p )( x − β p ) , for a fixed prime p ∤ N where u n ( α p , β p ) := a ( p n − ) = α np − β np α p − β p , and | a ( p ) | ≤ p k − .2.2. Modular Forms and their Galois Representation.Definition 2.5.
We say that a newform f ∈ S newk (Γ ( N ) , χ ) has trivial residual mod 2 Galoisrepresentation if a ( p ) is even for all p ∤ · N . Remark.
The condition p = 2 comes from the fact that the determinant of the representation ofthe Galois group evaluated at the Frobenius element needs to be nonzero in order to be invertible.Using Proposition . -(2), this implies that we can derive a ( p d ) to be odd if and only if d iseven. Similarly, we get that a ( p d ) is even if and only if d is odd. It follows that a ( n ) is odd ifand only if n is an odd square. Furthermore, requiring f ∈ S new (Γ ( N )) to have trivial residualmod Galois representation is equivalent to asking that the associated modular elliptic curvehas a rational -torsion point. Newforms of weight k = 2We begin by studying newforms of weight k = 2, level N and trivial character χ with integercoefficients. These modular forms are interesting by themselves as they correspond to modularelliptic 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 ∤ · N . The only defective oddvalues u d ( α p , β p ) are given in Table 1 by ( d, A, B, k ) ∈ { (3 , , , , (5 , , , } , and in rows 1 and 2 of Table 2. Remark.
For u ( α p , β p ) = ε r , we have ∤ a ( p ) and so u ( α, β ) is the first occurrence of inthe sequence. If | a ( p ) however, then ε r is no longer a defective value. SHORT NOTE ON INADMISSIBLE COEFFICIENTS OF WEIGHT 2 AND 2 k + 1 NEWFORMS 5 We will make use of the two following fundamental results.
Theorem 3.1 (Modularity Theorem) . Let E/ Q be an elliptic curve with conductor N anda rational -torsion point. Denote the associated newform with trivial residual mod Galoisrepresentation by f E ( z ) = P n ≥ a ( n ) q n ∈ S New (Γ ( N )) ∩ Z [[ q ]] . Then for all primes p ∤ · N of good reduction, we have a ( p ) = p + 1 − E ( F p ) , where E ( F p ) denotes the number of 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/ Q . Theorem 3.2 (Mazur’s Theorem) . If E/ Q is an elliptic curve, then E tor ( Q ) ∈ { Z /N Z | ≤ N ≤
10 or N = 12 } ∪ { Z / Z × Z /N Z | N = 2 , , , } . Furthermore, recall that if E/ Q has good reduction at p ∤ m for some m ∈ N , then thereduction map(3.1) π : E ( Q ) → E ( F p ) , is injective when restricted to m -torsion [20]. As a consequence, we get the following result. Lemma 3.2 (Lemma 3.1 [2]) . Suppose that E/ Q is an elliptic curve and that r | E tor ( Q ) .Then for all primes p ∤ · r · N , we have a ( p d ) ≡ p + p + · · · + p d mod r. Lemma 3.3 (Lemma 3.2 [2]) . If E/ Q has a rational and r -torsion point where r = 3 , , thenfor all gcd( n, · r · N ) = 1 , we have | a ( n ) | 6 = 1 . Integer points on Thue equations.
We discuss the general approach to solve the equa-tion a ( n ) = ℓ for some prime ℓ , positive or negative, where a ( n ) is the Fourier coefficient of f ∈ S newk (Γ ( N ) , χ ) ∩ Z [[ q ]]. Assume that | a ( n ) | 6 = 1 for some given values of n . Using Propo-sition 2 . (1)-(2) , we can see that studying a ( n ) = ℓ is equivalent to studying a ( p d − ) = ℓ for d (cid:12)(cid:12) | ℓ | ( | ℓ | − | ℓ | + 1). From the two-term recurrence relation satisfied by a ( p d − ), a ( p d − ) = ℓ 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 ) ∈ Z [ X, Y ] is a homogeneous polynomial, is called a
Thue equation. We will considerthose equations arising from the series expansion of(3.2) 11 − √
Y T + XT = ∞ X m =0 F m ( X, Y ) · T m = 1 + √ Y · T + ( Y − X ) · T + . . . Lemma 3.4. If a ( n ) satisfies Proposition . , and p ∤ N is a prime, then F m (cid:0) χ ( p ) p k − , a ( p ) (cid:1) = a ( p m ) . Hence, solving a ( p m ) = q boils down to computing integer solutions to the equation F m ( X, Y ) = ℓ. MALIK AMIR AND ANDREAS HATZIILIOU
Methods for solving Thue equations are implemented in Sage [5], Magma, and are bestsuited for m ≥
3. For m = 1 ,
2, these equations often have infinitely many solutions as theydo not represent curves with positive genus when the weight is k = 1 ,
2, hence we require extrainformation to infer their finiteness. In the case of weight 2 newforms, the idea is to introducea 3 and 5-torsion point to get additional congruences to avoid having to deal with infinitelymany solutions for the equations a ( p ) = ℓ, a ( p ) = ℓ . Indeed, note that d (cid:12)(cid:12) | ℓ | ( | ℓ | − | ℓ | + 1) ≡ , for all ℓ and hence d = 3 will always have to be checked.3.2. Some Congruences.
We now list congruences obtained using Lemma 3 . Lemma 3.5.
Let E/ Q be an elliptic curve with conductor N having a rational and -torsionpoint. Consider primes p for which gcd( p, · · N ) = 1 .a. If a ( p d − ) = ℓ = ± , then ( p, d ) = (1[3] , , (2[3] , , (2[3] , . b. If a ( p d − ) = ℓ ≡ , then ( p, d ) = (1[3] , , (2[3] , odd ) . c. If a ( p d − ) = ℓ ≡ , then ( p, d ) = (1[3] , . Remark.
In point b, the last pair is problematic as d is always odd. Hence, it is not possibleto provide a general result in this case.Let E/ Q be an elliptic curve with conductor N having a rational and -torsion point. Considerprimes p for which gcd( p, · · N ) = 1 .a. If a ( p d − ) = ± , then ( p, d ) = (1[5] , , (3[5] , , (3[5] , , (4[5] , , (4[5] , , (4[5] , . b. If a ( p d − ) = ℓ ≡ , then ( p, d ) = (1[5] , , (2[5] , , (3[5] , , (4[5] , , (4[5] , . c. If a ( p d − ) = ℓ ≡ , then ( p, d ) = (1[5] , , (2[5] , . d. If a ( p d − ) = ℓ ≡ , then ( p, d ) = (1[5] , , (3[5] , , (2[5] , . e. If a ( p d − ) = ℓ ≡ , then ( p, d ) = (1[5] , , (3[5] , . Using the above congruences extensively as well as the relative divisibility property of Lucasnumbers, we get the following result.
SHORT NOTE ON INADMISSIBLE COEFFICIENTS OF WEIGHT 2 AND 2 k + 1 NEWFORMS 7 Theorem 3.3.
Suppose f ( z ) = q + P n ≥ a ( n ) q n ∈ S new (Γ ( N )) ∩ Z [[ q ]] has trivial residualmod 2 Galois representation, namely, E/ Q is an elliptic curve of conductor N with a rational -torsion point. Then the following are true.(1) If E/ Q has a rational -torsion point, then for n > and gcd( n, · · N ) = 1 , we havea. If a ( n ) = 7 , , , , , then n = p with p = 2[3] .b. If a ( n ) = 29 then n = p d − = 13 and a ( p ) = ± .c. If a ( n ) = 41 then n = p d − = 43 and a ( p ) = ± .d. If a ( n ) = − then n = p d − = 7 and a ( p ) = ± .e. If a ( n ) = − then n = p d − = 7 and a ( p ) = ± .f. If a ( n ) = − then n = p d − = 167 and a ( p ) = ± .Furthermore, a ( n )
6∈ {− , , , − , , − , , , − , − , , , , − , − , , − , , , − } . (2) If E/ Q has a rational -torsion point, then for n > and gcd( n, · · N ) = 1 , we havea. If ℓ ≡ and a ( n ) = ℓ , then n = p and p ≡ .b. If ℓ ≡ , ℓ = − and a ( n ) = ℓ , then n = p and p ≡ .c. If ℓ ≡ , ℓ = 3 and a ( n ) = ℓ , then n = p and p ≡ , .Furthermore, a ( n )
6∈ {− , , − , , , − , − , , − , − , , , − } . Remark. In (1) , we have omitted the primes , , , , , , which are of the form ℓ ≡ ,and the primes − ℓ ≡ due to the large number of curves involved. However, following themethods outlined in this text, the interested reader will have no difficulty investigating thesecases. In (2) , note that the primes of the form ℓ ≡ are those of the outlined list. We’vealso ruled out − simply because it is a pretty number and a nice example of application ofTheorem 3.5. Theorem 3.4.
Let E/ Q be an elliptic curve of conductor N with a and -torsion point. Let n > and gcd ( n, · · N ) = 1 . If ℓ ≡ , ℓ = 5 and the odd prime divisors d of | ℓ | ( | ℓ |− | ℓ | +1) are not congruent to , then a ( n ) = ℓ . Theorem 3.5.
Let E/ Q be an elliptic curve of conductor N with a and -torsion point. Let n > and gcd( n, · · N ) = 1 .1. If ℓ ≡ and the odd prime divisors d of | ℓ | ( | ℓ | − | ℓ | + 1) are not congruent to , ,then a ( n ) = ℓ .2. If ℓ ≡ , ℓ = − and the odd prime divisors d of | ℓ | ( | ℓ | − | ℓ | + 1) are not congruentto , , then a ( n ) = ℓ .3. If ℓ ≡ , ℓ = 3 and the odd prime divisors d of | ℓ | ( | ℓ | − | ℓ | + 1) are not congruentto , , then a ( n ) = ℓ .4. If ℓ ≡ and the odd prime divisors d of | ℓ | ( | ℓ | − | ℓ | + 1) are not congruent to , ,then a ( n ) = ℓ . The above theorems can be made independent of the level using the following lemma.
MALIK AMIR AND ANDREAS HATZIILIOU
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 − ) = ( ( ± m if ord p ( N ) = 1 , p ( N ) ≥ . Theorem 3.6.
Let E/ Q be an elliptic curve with conductor N and f ( z ) the correspondingnewform with Fourier coefficients a ( n ) . For r = 3 , , suppose that · r divides | E tor ( Q ) | . Then a ( p d − ) = r v unless d = r .Proof. Let r = 3. Then by Lemma 3.5, 3 (cid:12)(cid:12) | a ( p d − ) | if and only if 3 | d . Indeed, if p ≡ , a ( p d − ) ≡ p ≡ | d . Suppose that a ( p d − ) = 3 v and d >
3, then a ( p d − ) is also a multiple of 3, which contradicts that 3 v is not defective. Thus d = 3 is theonly solution.Let r = 5. Then by Lemma 3.5, 5 (cid:12)(cid:12) | a ( p d − ) | if and only if p ≡ | d . For d >
5, wehave a ( p d − ) is also a multiple of 5, violating that a ( p d − ) is not defective. Hence d = 5. (cid:3) Newforms of odd weight k ≥ ± ℓ as Fouriercoefficients of odd weight k ≥ N andnebentypus χ given by a real quadratic Dirichlet character and trivial residual mod 2 Galoisrepresentation. Theorem 4.1.
Let gcd( n, · N ) = 1 . Then a ( p d − ) = ± and for n > we have a ( n ) = ± .It follows that if a ( n ) = ± ℓ for some prime ℓ , then n = p d − where d | ℓ ( ℓ − is odd. If ± ℓ isnot defective, then d is an odd prime.Proof. Note that ± a ( p d − ) = −
1, then it must be at u = a ( p ) = −
1, where the constraints imply thatwe must satisfy χ ( p ) p k − = a ( p ) + 1. Since k − k − m . Then we’releft with the equation χ ( p ) x m = y + 1 and clearly there are no solutions for χ ( p ) = 0 , −
1. If χ ( p ) = 1 then we obtain ( x m ) − y = 1 and this gives us the integer solutions x = ± , y = 0which aren’t allowed. Hence a ( p d − ) = − p, · N ) = 1. Now for row 2, if a ( p d − ) = 1then it must happen for u = a ( p ) = 1 with constraints given by χ ( p ) p k − = a ( p ) − a ( p ) > χ ( p ) = 0 , − ,
1. Hence a ( p d − ) = ± (cid:3) Lemma 4.1.
The curve a ( p ) = ± ℓ has no solutions if χ ( p ) = 0 . If χ ( p ) = 1 then the curvehas the form ( y − x m )( y + x m ) = ± ℓ which has a unique solution depending on ℓ only. For χ ( p ) = − the curve has the form y + ( x m ) = ± ℓ which has no solutions for − ℓ and hasfinitely many solutions for + ℓ as it is a sum of squares. Hence, to rule out or locate any odd prime value ℓ as a Fourier coefficient of f ( z ), it sufficesto follow the following steps. Let gcd( n, · N ) = 1 and ℓ be an odd prime.Step 1. By multiplicativity of the Fourier coefficients, we have that a ( n ) = ± ℓ if and only if Y i a ( p d i − i ) = ± ℓ. SHORT NOTE ON INADMISSIBLE COEFFICIENTS OF WEIGHT 2 AND 2 k + 1 NEWFORMS 9 Step 2. Use Theorem 4.1.Step 3. Use the Sage Thue solver to solve a ( p d − ) = ± ℓ and analyze the solutions. Lemma 4.2 (Proposition 13.3.14 [9]) . Let p | N and assume that χ can be defined modulo N/p and let f = P a ( n ) q n ∈ S newk (Γ ( N ) , χ ) be a normalized eigenform. Then1. If p | N , then a ( p ) = 0 .2. If p ∤ N , then a ( p ) = χ ( p ) p k − where χ is the character modulo N/p equivalent to χ . Using this lemma, we easily obtain the following.
Corollary 4.2.
Suppose that χ can be reduced modulo N/p and call it χ . Then a ( p m ) = 0 forall m ≥ and a ( n ) = 0 for all ( n, N ) ≥ p .Proof. Recall that a ( p m ) = a ( p ) a ( p m − ) and if p | N the result follows immediately. For p ∤ N we have a ( p ) = χ ( p ) p k − . Since k is odd, k − χ ( p ) = 0 then a ( p ) cannot be aninteger which is what we require for our newforms. Hence a ( p ) = 0 is the only possibility. (cid:3) Lemma 4.3 (Corollary 13.3.17 [9]) . Let p | N and assume that χ cannot be defined modulo N/p .If f ∈ S newk (Γ ( N ) , χ ) is a normalized eigenform, then | a ( p ) | = p k − . Corollary 4.3.
Let p | N and assume that χ cannot be defined modulo N/p . Then a ( p m ) =( ± m ( p k − ) m . Newforms of weight k = 1In 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 mayhave that n | d and u n = −
1. From the first row of the defective table, we know that this mayhappen only if n = 3, namely if u = a ( p ) = −
1. Thus, we would like to avoid 3 | d but thatis not possible as ℓ ( ℓ − ℓ + 1) is always divisible by 3. This implies that we only have theinformation that d | ℓ ( ℓ − ℓ + 1) is odd and no longer an odd prime. First, this leads us to alarge 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 = ± ℓ + χ ( p )and is incredibly difficult to solve as it is still an open problem to determine if there are infinitelymany primes of the form ℓ = y + 1. 6. Appendix ( A, B ) Defective u n ( α, β )( ± , ) u = − u = 7, u = ∓ u = ± u = − u = ± u = ∓ ± , ) u = 1, u = ± ± , ) u = 1, u = ∓ ± , ) u = 1, u = ∓ ± , ) u = ∓ ± , ) u = 5( ± , ) u = ± ± , ) u = ∓ ± , ) u = 1( ± , ) u = ± Table 1.
Sporadic family of defective u n ( α, β ) satisfying equation 2.2 in evenweight k including k = 2 [4] . SHORT NOTE ON INADMISSIBLE COEFFICIENTS OF WEIGHT 2 AND 2 k + 1 NEWFORMS 11 ( A, B ) Defective u n ( α, β )( ± , − u = 5 , u = ± ± , ) u = − u = 7, u = ∓ u = ± u = − u = ± u = ∓ ± , ) u = 5, u = ∓ ± , ) u = 1, u = ± ± , ) u = 1, u = ∓ ± , ) u = ∓ ± , ) u = ∓ ± , ) u = 5( ± , ) u = ∓ Table 2.
Sporadic family of defective u n ( α, β ) satisfying equation 2.2 in oddweight k ≥ . ( A, B ) Defective u n ( α, β ) Constraints on parameters( ± m, p ) u = − m > p = m + 1( ± m, p k − ) u = ε r ( p, ± m ) ∈ B r,ε ,k with 3 ∤ m, ( ε, r, m ) = (1 , , , and m ≥ ε r − ( ± m, p k − ) u = ∓ m ( p, ± m ) ∈ B ,k with m > ± m, p k − ) u = ± εm ( p, ± m ) ∈ B ε ,k with ( ε, m ) = (1 , m > ± m, p k − ) u = ± ( − r m (2 m + ( − r ) / p, ± m ) ∈ B r ,k with gcd( m,
6) = 1 , ( r, m ) = (1 , , and m ≥ ( − r +2 ( ± m, p k − ) u = ± εm (2 m + 3 ε ) ( p, ± m ) ∈ B ε ,k with 3 | m and m > ± m, p k − ) u = ± r +1 εm ( m + 3 ε · r − ) ( p, ± m ) ∈ B r,ε ,k with m ≡ m ≥ ε · r +2 Table 3.
Parameterized family of defective u n ( α, β ) satisfying equation 2.2 ineven weight 2 k ≥ Notation: m, k, r ∈ Z + , ε = ± , p is a prime number. SHORT NOTE ON INADMISSIBLE COEFFICIENTS OF WEIGHT 2 AND 2 k + 1 NEWFORMS 13 ( A, B ) Defective u n ( α, β ) Constraints on parameters( ± m, χ ( p ) p k − ) u = − χ ( p ) p k − = m + 1( ± m, χ ( p ) p k − ) u = 1 χ ( p ) p k − = m − m > ± m, χ ( p ) p k − ) u = ε r χ ( p ) p k − = m − ε r with 3 ∤ m, ( ε, r, m ) = (1 , , , and r > ± m, χ ( p ) p k − ) u = ± εm χ ( p ) p k − = m − ε with 2 ∤ m, m = 1( ± m, χ ( p ) p k − ) u = ± εm χ ( p ) p k − = m − ε with 2 | m, ( ε, m ) = (1 , ± m, χ ( p ) p k − ) u = ( ± m ± m ) / χ ( p ) p k − = m − ∤ m > ± m, χ ( p ) p k − ) u = ± εm ± m χ ( p ) p k − = m − ε with 3 | m ( ± m, χ ( p ) p k − ) u = ( ± m ( − r ± m (( − r ) ) / χ ( p ) p k − = 4 m − ( − r +2 r > , m ≡ ± , ( r, m ) = (1 , ± m, χ ( p ) p k − ) u = ± m ε · r ± m (2 r ) χ ( p ) p k − = 4 m − · r +2 ε, r > , m ≡ Table 4.
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