COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS. Jan H. Bruinier and Ken Ono. Appearing in the Journal of Number Theory

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1 COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS Jan H. Bruinier and Ken Ono Aearing in the Journal of Number Theory Abstract. In this aer we study the distribution of the coefficients an of half integral weight modular forms modulo odd integers M. As a consequence we obtain imrovements of indivisibility results for the central critical values of quadratic twists of L-functions associated with integral weight newforms established in [O-S]. Moreover, we find a simle criterion for roving cases of Newman s conjecture for the artition function. 1. Introduction and Statement of Results Suose that w 1 Z, that N is a ositive integer with 4 N if w Z, and that χ is a Dirichlet character whose conductor divides N. Let S w N, χ denote the sace of weight w cus forms with resect to the congruence subgrou Γ 0 N with Nebentyus character χ [K, Sh] are standard references. As usual, we shall identify every such cus form fz with its Fourier exansion where q = e πiz throughout fz = anq n. Insired by Kolyvagin s work on the Birch and Swinnerton-Dyer Conjecture and works of Kohnen, Zagier and Waldsurger relating the coefficients of half-integral weight Hecke eigenforms to values of modular L-functions, there have been a number of works on the indivisiblity of the coefficients of half-integral weight cus forms. For examle, works by Bruinier, Jochnowitz, McGraw, and Ono and Skinner [Br, J, M, O-S] imly that if fz = anqn S λ+ 1 N, χ Z[[q]] is an eigenform which is not a single variable theta series, then every sufficiently large rime l has the roerty that there are infinitely many 1991 Mathematics Subject Classification. Primary 11F33 ; Secondary 11P83. Key words and hrases. half-integral weight modular forms, the artition function. The second author is grateful for the suort of the Alfred P. Sloan, David and Lucile Packard, and H. I. Romnes Fellowshis. Both authors thank the Number Theory Foundation and the National Science Foundation for their suort. 1 Tyeset by AMS-TEX

2 JAN H. BRUINIER AND KEN ONO square-free integers n for which an 0 mod l. Although this result is satisfying, many questions remain. For examle, it is natural to ask for a recise and natural arithmetic descrition of these rimes l. Much more is known about the coefficients of integer weight cus forms. The arithmetic of Galois reresentations and the combinatorics of Hecke oerators dictate their behavior. For examle, these arguments are very useful for studying the distribution of the coefficients modulo M. Using Galois reresentations and a delightfully simle argument, Serre observed [6.4, S] that there is a set of rimes with ositive density with the roerty that 1.1 an r r + 1an mod M for every air of ositive integers r and n. Obviously, 1.1 imlies that each residue class modulo M contains infinitely many coefficients rovided that there is an n for which gcdan, M = 1. Half-integral weight cus forms do not necessarily enjoy this roerty. To see this, notice that Dedekind s function η4z := q 1 q4n S 1 576, χ 1 here χ 1 = 1 has the q-exansion η4z = χ 1 nq n = q q 5 q 49 + q 11 + q 169. We begin by determining conditions which guarantee that a half-integral weight cus form ossesses this roerty modulo an odd integer M. Theorem 1. Let fz = anqn S λ+ 1 N, χ Z[[q]] be a half-integral weight cus form, and let χ be a real Dirichlet character. If M is an odd integer and there is a ositive integer n for which gcdan, M = 1, then at least one of the following is true: 1 If 0 r < M, then #{0 n X : an r mod M} r,m { X/ log X if 1 r < M, X if r = 0. There are finitely many square-free integers, say n 1, n,..., n t, for which fz t an i m q n im i=1 m=1 Moreover if gcdm, N = 1, ɛ {±1} and NM is a rime with n i {0, ɛ} for each 1 i t, then 1fz is an eigenform modulo M of the half-integral weight Hecke oerator T, λ, χ. In articular, we have 1 1fz T λ, λ, χ ɛχ λ + λ 1 1fz

3 Remarks. COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS 3 1 For simlicity, the results here are stated for cus forms with integer coefficients and real Nebentyus character. However, we stress that Theorem 1, and Corollaries and 3 aly for any half-integral weight cus form with algebraic integer coefficients. Theorem 1 1 requires a minor modification. If α is a suitable algebraic integer and M is a suitable ideal, then one obtains the frequency that an rα mod M, for every odd number r. In view of the single variable theta series and those forms congruent to such series, it turns out that the estimates in Theorem 1 1 are nearly otimal. However, aart from such forms, it is lausible that each residue class r contains a ositive roortion of an 3 Conclusions 1 and in Theorem 1 are not necessarily mutually exclusive. In fact, one may often emloy Theorem 1 to rove Theorem 1 1 see Theorem 4. 4 Suose that fz is a Hecke eigenform which is not a single variable theta series. If fz satisfies Theorem 1 and gcd 1, M = 1, then Deligne s theorem bounding Hecke eigenvalues requires that M λ 1. Theorem 1 has a variety of number theoretic alications, and we begin with the arithmetic of the Shimura corresondence. If fz = anqn S λ+ 1 N, χ Z[[q]] is a Hecke eigenform, then we address the roblem described at the outset i.e. that of obtaining a recise and urely arithmetic descrition of those rimes l for which there are only finitely many square-free n with an 0 mod l. To motivate our result, we recall an examle of Kohnen and Zagier [K-Z]. If z = τnqn = q 4q + S 1 1, χ 0, then Kohnen and Zagier roved that the function f z S 13/ 4, χ 0 defined by f z = anq n = 60 πi G 44zΘ z G 44zΘz = q 56q q 5... throughout χ 0 denotes the trivial character is a reimage of z under the Shimura corresondence. Here G 4 z is the usual weight 4 Eisenstein series on SL Z and Θz = 1 + qn. It turns out that 1. f z = anq n n q n mod 5. 5 Obviously n = 1 is the only square-free integer for which an 0 mod 5. Ramanujan roved that if is rime, then 1.3 τ + mod 5. In view of Theorem 1, it is natural to susect a strong relationshi between congruences 1. and 1.3.

4 4 JAN H. BRUINIER AND KEN ONO In the late 1960s and early 1970s, Serre and Swinnerton-Dyer [SwD] emloyed Deligne s theory of Galois reresentations to exlain congruences such as 1.3. Suose that F z = Anqn S λ N 1, χ 0 Z[[q]] is a normalized Hecke eigenform. If l is rime, then Deligne roved that there is a Galois reresentation 1.4 ρ l,f : GalQ/Q GL Z/lZ such that for every rime N 1 l we have Trρ l,f Frob A mod l, detρ l,f Frob λ 1 mod l. A rime l 5 is called excetional if Imρ l,f GalQ/Q does not contain SL Z/lZ. The Serre and Swinnerton-Dyer theory [SwD, R1, R, R3] imlies that congruences like 1.3 hold recisely for excetional rimes l. Combining these ideas with Theorem 1, we obtain: Corollary. Suose that fz = anqn S λ+ 1 N 0, χ Z[[q]] is an eigenform that is a reimage of a newform F z = Anqn S λ N 1, χ 0 Z[[q]] under the Shimura corresondence. 1 If l 5 is a non-excetional rime for which l N 0 and fz 0 mod l, then there are infinitely many square-free integers n for which an 0 mod l. If F z has comlex multilication and l N 0 is a rime for which fz 0 mod l, then there are infinitely many square-free integers n for which an 0 mod l. Remark. Every F z without comlex multilication has at most finitely many excetional rimes l, and they are easily determined see [SwD], [Th..1, R3]. By Kolyvagin s celebrated work on the Birch and Swinnerton-Dyer Conjecture, it is well known that results like Corollary have many consequences for ellitic curves. For examle, recent similar works [Br, J, O-S] contain, for sufficiently large rimes l, results regarding the frequency of quadratic twists of ellitic curves with analytic rank 0 whose Tate-Shafarevich grous lack l-torsion, as well as effective uer bounds for the order of the l-art of the Tate-Shafarevich grou of ellitic curves with analytic rank 1. Corollary in the resent work yields more recise versions of these results by clarifying what is meant for a rime l to be sufficiently large. Since the consequences i.e. [Cor. -5, O-S] follow from [Cor. 1, O-S] in a straightforward way, here we content ourselves by stating its imrovement. We begin with some notation. Suose that F z = Anqn S k N, χ 0 is an even integer weight newform. If D is the fundamental discriminant of a quadratic field which is corime to N, then let F χ D z denote the quadratic twist of F z defined by 1.5 F χ D z = D Anq n. n

5 COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS 5 Moreover, let D 0 be defined by { D if D is odd, 1.6 D 0 := D /4 if D is even. A non-zero comlex number Ω C is a nice eriod for F z if 1.7 LF χ D, kd k 1 0 Ω, ɛd > 0, is always an algebraic integer. Here LF χ D, s denotes the L-function of F χ D z, and ɛ {±1} denotes the sign of the functional equation of LF, s, the L-function of F z. If l is rime and l denotes the usual multilicative valuation at l extended to the algebraic closure of Q, then we obtain the following imrovement of [Cor. 1, O-S]: Corollary 3. Let F z = Anqn S k N, χ 0 Z[[q]] be an even integer weight newform, and let ɛ be the sign of the functional equation for LF, s. There is a nice eriod Ω for F z with the roerty that every non-excetional rime l 5 with l N has infinitely many fundamental discriminants D for which ɛd > 0 and LF χ D, kd k 1 = 1. Ω l Theorem 1 also alies to a classical conjecture in additive number theory. A artition of a ositive integer n is any non-increasing sequence of ositive integers whose sum is n. Let n denote the number of artitions of n as usual, we adot the convention that 0 = 1 and α = 0 if α N. If l 5 is rime, then define 1 γ l < 4 by the condition 1.8 4γ l 1 mod l. If l = 5, 7 or 11, then Ramanujan roved for every non-negative integer n that 1.9 ln + γ l 0 mod l. Recently we have learned that similar, but more comlicated congruences, are quite common see [A, O]. For examle if M is corime to 6, then there are integers A and B such that for every n we have An + B 0 The congruence n mod 13 is a tyical examle. Although there are many congruences, numerical evidence suggests that congruences of the secial form 1.9 only hold for l = 5, 7 and 11.

6 6 JAN H. BRUINIER AND KEN ONO Conjecture R. If l 13 is rime, then there are infinitely many integers n for which ln + γ l 0 mod l. The following classical conjecture of Newman [N] concerns the distribution of the artition function among the comlete set of residue classes modulo an integer M. Conjecture N. Newman If M is a ositive integer, then for every integer 0 r < M there are infinitely many non-negative integers n for which n r Works by Atkin, Kolberg and Newman [At, Ko, N] verified the conjecture for M =, 5, 7 and 13 note: the M = 11 case follows similarly. More recently, the second author and Ahlgren [A, O] obtained an algorithm which resumably roves the truth of the conjecture for any given M corime to 6. Theorem 1 leads to an interesting connection between Conjectures R and N, one which roduces a simler algorithm for testing Newman s Conjecture for rime moduli. Theorem 4. If l 5 is rime, then at least one of the following is true: 1 Newman s Conjecture is true for M = l, and #{0 n X : n r mod l} r,l { X/ log X if 1 r < l, X if r = 0. For every integer n we have ln + γ l 0 mod l. In view of this result, Newman s Conjecture for a rime modulus M = l 5 follows from the existence of a single n for which ln + γ l 0 mod l. Corollary 5. Conjectures N and R are true for every rime 13 M < The roof of Theorem 1 requires Shimura s theory of half-integral weight modular forms, a result of Serre on the coefficients of integer weight cus forms, and some commutation relations for half-integral weight Hecke oerators. In we make some reliminary reductions for the roof of Theorem 1, and in 3 we conclude the roof with an analysis of the action of the half-integral weight Hecke oerators modulo M. In 4 we rove Theorem 4 and Corollaries, 3 and 5.. Preliminary reductions Throughout this section let M denote an odd integer, and let.1 fz = anq n S λ+ 1 N, χ Z[[q]]

7 COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS 7 be a half-integral weight cus form with integer coefficients and real Dirichlet character χ. If N is rime, then the half-integral weight Hecke oerator T, λ, χ is a linear endomorhism on the sace S λ+ 1 N, χ which is defined by. fz T, λ, χ := a n + χ n Here χ is the Dirichlet defined by χ n := 1 λ χn. n λ 1 an + χ λ 1 an/ q n. Lemma.1. Suose that M is an odd integer and 0 1 mod NM is a rime for which fz T, λ, χ fz If there is a ositive integer n 0 for which n 0 0 = 1 and gcdan0, M = 1, then for every 0 r < M there are infinitely many integers n with an r Proof. By hyothesis, we see that χ 0 = 1 and 0 1 mod 4. Therefore. imlies, for every ositive integer n, that.3 an 0 Therefore we find that n 0 an an/ 0.4 an 0 0 3an 0 Since n 0 = 0 if 0 n,.3 and.4 imly that Generally, if k is a ositive integer, then an an 0 0 an 0 5an 0 mod M, an an an 0 7an 0 mod M, an 0 k 0 k + 1an 0 Since gcdan 0, M = 1, the result follows by varying k. We now turn to the existence of such rimes 0. The next result, which follows from an observation of Serre and the arithmetic of the Shimura corresondence, roves that there is a vast suly of such rimes.

8 8 JAN H. BRUINIER AND KEN ONO Lemma.. A ositive roortion of the rimes 1 mod NM have the roerty that fz T, λ, χ fz Proof. By relacing fz by any congruent form modulo M, we may assume that fz is not a linear combination of single variable theta series. Hence its image, say F z, under the Shimura corresondence see [K, Sh] is an even integral weight cus form in the sace S λ N, χ 0 Z[[q]]. Serre observed [6.4, S] that a subset of rimes 1 mod NM with ositive density have the roerty that.5 F z T λ, χ 0 F z Here T λ, χ 0 denotes the usual th Hecke oerator on the sace S λ N, χ 0. Denote this set of rimes by Sf, M. Since the Shimura corresondence commutes with the Hecke oerators for the saces S λ+ 1 N, χ and S λn, χ 0, if Sf, M, then.5 imlies that fz T, λ, χ 1 fz Using Lemma. and Lemma.1, we now make an imortant observation. Theorem.3. If there is a ositive integer n for which gcdan, M = 1, then at least one of the following is true: 1 If 0 r < M, then there are infinitely many integers n for which an r There are finitely many square-free integers, say n 1 < n < < n t, for which t fz an i m q n im i=1 m=1 Proof. As in the roof of Lemma., let Sf, M denote the set of rimes 1 mod NM for which fz T, λ, χ fz Suose that 1 is false. If Sf, M, then Lemma.1 imlies that every n Z + with gcdan, M = 1 has the roerty that n.6 {0, 1}. Let n 1 < n <... denote the sequence of square-free ositive integers with the roerty there is an integer m i for which an i m i 0 By.6, each n i has the roerty that n i {0, 1} for every rime Sf, M. By quadratic recirocity, Sf, M cannot contain a ositive roortion of the rime numbers if there are infinitely many such n i s. Therefore there are finitely many square-free integers, say n 1 < n < < n t, such that t fz an i m q n im i=1 m=1

9 COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS 9 3. Hecke eigenvalues modulo M and the roof of Theorem 1 Here we consider the arithmetic of those modular forms fz satisfying Theorem.3. To do so, we rove a general statement regarding the eigenvalues of the half-integral weight Hecke oerators modulo M. As in the last section, throughout we assume that M is an odd integer, and that 3.1 fz = anq n S λ+ 1 N, χ Z[[q]] is a half-integral weight cus form with integer coefficients and real character χ. First we recall some oerators on the sace S λ+ 1 N, χ see [S-St, Br]. The Fricke involution W N : S λ+ 1 N, χ S λ+ 1 N, N χ is defined by 3. fz W N = i Nz λ 1/ f 1/Nz. If m is a ositive integer, then let B m : S λ+ 1 N, χ S λ+ 1 Nm, χ be the rojection defined by 3.3 fz B m = amnq mn. Finally, if ψ is a Dirichlet character with conductor m and dz = cnqn S λ+ 1 N, χ, then let d ψ z S λ+ 1 Nm, χψ denote the twist of dz by ψ: 3.4 d ψ z = ψncnq n. These oerators satisfy various commutation relations see [S-St], [Br]. For instance, if N is rime, then the twist by the quadratic character ϕ = and the Fricke involution W N satisfy [3, Br] 3.5 f ϕ W N = χ 1/ f W N B 1/ f W N. Theorem 3.1. Let M be a ositive integer that is corime to N, and let NM be rime. If there is an ɛ {±1} for which fz n {0,ɛ} anq n mod M, then 1fz T, λ, χ ɛχ λ + λ 1 1fz

10 10 JAN H. BRUINIER AND KEN ONO Proof of Theorem 3.1. We argue in a similar way as in the roof of [Th. 1, Br]. Without loss of generality we may assume that N is a square. Let gz S λ+ 1 N, χ be the cus form defined by 3.6 gz = bnq n = fz W N. Let ϕ = and define 3.7 hz = f f B ɛf ϕ = ϕn= ɛ anq n. By hyothesis, we have that hz 0 We consider the twist of gz with the Dirichlet character ϕ. The commutation relation 3.5 imlies that g ϕ W N = χ = χ 1/ f B 1/ f 1/ 1/ f ɛχ 1/ f ϕ χ 1/ h. If we aly W N once again and use 3.5 for f ϕ z, we get g ϕ = χ 1/ 1/ f W N + ɛg ɛg B χ 1/ h W N. We substitute f W N z = λ+1/ f W N z = λ+1/ g z and obtain the ower series identity ϕnbnq n = χ λ+1 λ bnq n ɛ 1 bnq n + ɛ gcdn,=1 bnq n χ 1/ h W N. Since fz has coefficients in Z, the q-exansion rincile on the modular curve X 0 N imlies that the coefficients bn of gz are contained in Z[1/N, ζ N ]. Here ζ N denotes a rimitive N-th root of unity. Because hz 0 mod M, we find in the same way that the coefficients of h W N are contained in the rincial ideal MA of the ring A := Z[1/N, ζ N ] see also [Lemma 1, Br]. Notice that the assumtion gcdm, N = 1 is needed here. For b A we write b 0 mod M, if b MA. From the identity 3.7 we obtain the following congruences for the coefficients bn modulo M: 1 If does not divide n, then bn ɛϕnbn If n and does not divide n, then 1bn 0 3 If n, then 1bn ɛχ λ 1bn/

11 COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS 11 Inserting these congruences into., the formula for gz T, λ, χ, we find that 1gz T, λ, χ ɛχ λ + λ 1 1gz The theorem now follows from the fact that the Fricke involution W N commutes with the Hecke oerator T, λ, χ. Proof of Theorem 1. In view of Theorems.3 and 3.1, it suffices to rove the estimates in Theorem 1 1. By Theorem.3, for each 0 r < M, there is a ositive integer n r for which 3.8 an r r Obviously, we have fz S λ+ 1 N r n r, χ. Hence arguing as before, [6.4, S] and the commutativity of Shimura s corresondence imlies that a ositive roortion of the rimes 1 mod MN r n r have the roerty that fz T, λ, χ 0 Call this set of rimes Zf, M. If Zf, M, then. imlies for each n r that 3.9 a n r 1 λ χ nr 1 r Suose that n r has the rime factorization n r = er i i,r, where each i,r is odd. Since every Zf, M satisfies 1 mod 8 and 1 mod i,r, by quadratic recirocity we have er er nr = i i,r = Therefore for every Zf, M, 3.9 imlies that a n r 1 λ χ 1r i i,r 1 i,r For each these values constitute a comlete set of reresentatives for the residue classes modulo M. By varying, we find that = i #{0 n X : an r mod M} X/ log X. 1 i,r = 1. For the r = 0 estimate, notice that if Zf, M, then for all n. imlies that n a n χ λ 1 an λ 1 an/ By relacing n by n where n, this becomes a 3 n λ 1 an/ 0 This immediately imlies that a roortion of n have an 0

12 1 JAN H. BRUINIER AND KEN ONO 4. Number Theoretic Alications Here we consider the number theoretic consequences of Theorem 1 described in 1. Proof of Corollary. Begin by observing that if fz is an eigenform, then. imlies that for square-free n we have an anm for all m. Therefore, if anm 0 mod l, then an 0 mod l. Suose that there are finitely many square-free integers, say n 1, n,..., n t, for which fz t i=1 m=1 an i m q n im mod l. By Theorem 1, there are arithmetic rogressions of rimes r j mod T N 0 l, for some ositive integer T, with the roerty that fz T, λ, χ ɛχ r j r λ j + r λ 1 j fz mod l for every rime r j mod l. Moreover, these residue classes r j mod T N 0 l cover at least l 3/ many residue classes modulo l note. this deends on whether l divides any n i. Since the coefficient A is the eigenvalue of fz with resect to the Hecke oerator T, λ, χ by the definition of Shimura s corresondence, for rimes r j mod T N 0 l we obtain the congruence 4.1 A ɛχ r j r λ j + r λ 1 j mod l. Case 1. If l is non-excetional, the Chebotarev Density Theorem, the uniform distribution of Frob in the Galois grou, and an an easy generalization of [Lemma 7, SwD] imlies that the A do not satisfy any congruences like 4.1. Therefore there are infinitely many square-free integers n with an 0 mod l. Case. Suose that F z has comlex multilication. By 4.1, it is easy to see that A 0 mod l only for those rimes above for which 1 mod l. However, since l N 0, every residue class r mod N 0 with gcdr, N 0 =1 contains some such rimes with 1 mod l. In articular, each class contains rimes for which A 0 mod l. However this is a contradiction; for if F z had comlex multilication, then there would be a discriminant D dividing N 0 with the roerty that A = 0 for every rime with D = 1. Hence there are infinitely many square-free n for which an 0 mod l. Proof of Corollary 3. There is a twist F χ, χ 1 = 1 k ɛ, satisfying Hyotheses H1 and H of [ , Wal]. By [Théorème 1, Wal] there is an integer N, a non-zero eigenform fz = anqn S k+ 1 N such that N N, and a eriod Ω such that for each fundamental discriminant D for which ɛd > 0 ad 0 = { LF χ ε D,kD k 1 0 D Ω if D 0 is relatively rime to 4N, 0, otherwise,

13 COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS 13 where ε D is an algebraic integer with ε D l = 1. The result now follows from Corollary by scaling Ω aroriately. Proof of Theorem 4. By [Th. 8, O], there is a half-integral weight cus form F l z = a lnq n S λl l, χ 1 Z[[q]] for which ln F l z q n mod l. n 0, ln 1 mod 4 Here λ l = l l / and χ 1 := 1. This is the k = 1 case of [Th. 8, O]. Although this theorem asserts that these forms are on the congruence subgrou Γ 0 576l, the roof shows that they are indeed on Γ To see this, observe that the Ul oerator modulo l is the lth Hecke oerator for modular forms on SL Z, and that η4lz η4z l mod l. Now observe that the coefficients of F l z mod l are recisely the values ln + γ l mod l. Moreover, observe that if a l n 0 mod l, then gcdn, 4 = 1. If Theorem 1 1 is false for F l z, then by Theorem 1 there are finitely many squarefree integers, say n 1, n,..., n t, for which 4 t 4.3 F l z a l n i m q n im i=1 m=1 Without loss of generality, we may assume that F l z a l n 1 m q n 1m m=1 mod l. mod l. This is easily accomlished by recursively relacing F l z by a suitable linear combination of trivial and quadratic twists. Fix an integer n 0 for which a l n 0 0 mod l. As before, we have that gcdn 0, 4 = 1. If P l denotes the set of rimes that are rimitive roots modulo l, then for all but finitely many rimes P l we may aly Theorem 1 with ɛ = n 0. If 0 n 0 is such a rime, then we have 4.5 a l k 0 n 0 a l 0n 0 n0 0 a l 4 0n 0 a l n 0 mod l. χ a l n 0 mod l, This follows from., Theorem 1, the fact 0 is a quadratic non-residue modulo l, and the fact that λ l = ll 1/ 1. More generally, for every ositive integer k we have { n0 0 χ 1 0 k 0 a ln 0 mod l if k is odd, k 0 a ln 0 mod l if k is even.

14 14 JAN H. BRUINIER AND KEN ONO Since gcdn 0, 4 = 1, we may select such a rime 0 with the additional roerty that n 0 0 χ 1 0 = 1. Since 0 is a rimitive root modulo l, 4.5 then imlies that each nonzero residue class r mod l contains infinitely many a l n. One obtains estimates in these cases by arguing as in the roof of Theorem 1. Similarly, one obtains the r 0 mod l case by arguing again in the roof of Theorem 1. Proof of Corollary 5. By Theorem 4, if there is a single n for which ln + γ l 0 mod l, then Newman s Conjecture is true for l. Moreover, since F l z mod l cannot be a non-zero olynomial, there must be infinitely many n for which ln + γ l 0 mod l. A simle rogram yields this result. References [A] S. Ahlgren, The artition function modulo comosite integers M, Math. Annalen , [At] A. O. L. Atkin, Multilicative congruence roerties and density roblems for n, Proc. London Math. Soc , [Br] J. H. Bruinier, Nonvanishing modulo l of Fourier coefficients of half integral weight modular forms, Duke Math. J , [J] N. Jochnowitz, Congruences between modular forms of half integral weights and imlications for class numbers and ellitic curves, rerint. [K] N. Koblitz, Introduction to ellitic curves and modular forms, Sringer Verlag, [K-Z] W. Kohnen and D. Zagier, Values of L series of modular forms at the center of the critical stri, Invent. Math , [Ko] O. Kolberg, Note on the arity of the artition function, Math. Scand , [M] W. McGraw, On a theorem of Ono and Skinner, J. Number Theory , [N] M. Newman, Periodicity modulo m and divisibility roerties of the artition function, Trans. Amer. Math. Soc , [O] K. Ono, Distribution of the artition function modulo m, Annals of Mathematics , [O-S] K. Ono and C. Skinner, Fourier coefficients of half-integral weight modular forms modulo l, Annals of Mathematics , [R1] K. Ribet, On l-adic reresentations attached to modular forms, Invent. Math , [R] K. Ribet, Galois reresentations attached to eigenforms with Nebentyus, Sringer Lect. Notes , [R3] K. Ribet, On l adic reresentations attached to modular forms II, Glasgow Math. J , [S] J.-P. Serre, Divisibilité de certaines fonctions arithmétiques, L Ensein. Math. 1976, [S-St] J.-P. Serre and H. Stark, Modular forms of weight 1/, Modular functions of one variable, VI, Sringer Lect. Notes , [Sh] G. Shimura, On modular forms of half-integral weight, Annals of Mathematics , [SwD] H. P. F. Swinnerton-Dyer, On l-adic reresentations and congruences for coefficients of modular forms, Sringer Lect. Notes , 1-55.

15 COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS 15 [Wal] J.-L. Waldsurger, Sur les coefficients de Fourier des formes modulaires de oids demi-entier, J. Math. Pures et Al , Deartment of Mathematics, University of Wisconsin, Madison, Wisconsin address: Deartment of Mathematics, University of Wisconsin, Madison, Wisconsin address: