REFINED DIMENSIONS OF CUSP FORMS, AND EQUIDISTRIBUTION AND BIAS OF SIGNS

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1 REFINED DIMENSIONS OF CUSP FORMS, AND EQUIDISTRIBUTION AND BIAS OF SIGNS KIMBALL MARTIN Abstract. We refine nown dimension formulas for saces of cus forms of squarefree level, determining the dimension of subsaces generated by newforms both with rescribed global root numbers and with rescribed local signs of Atin Lehner oerators. This yields recise results on the distribution of signs of global functional equations and sign atterns of Atin Lehner eigenvalues, refining and generalizing earlier results of Iwaniec, Luo and Sarna. In articular, we exhibit a strict bias towards root number +1 and a henomenon that sign atterns are biased in the weight but erfectly equidistributed in the level. Another consequence is lower bounds on the number of Galois orbits. Introduction Let S new (N denote the new subsace of weight ellitic cus forms on Γ 0 (N. Dimensions of such saces are well nown (cf. [Mar05]. If N > 1, one can decomose, in various ways, S new (N into certain natural subsaces. A crude decomosition is into the lus and minus saces S new,± (N, which are the subsaces of S new (N generated by newforms with global root number ±1, i.e., sign ± in the functional equation of their L-functions. A more refined decomosition is to consider subsaces generated by newforms with fixed local comonents π at rimes N, where each π is a reresentation of GL 2 (Q of conductor v(n. In this aer, we obtain exlicit dimension formulas for both of these tyes of subsaces in the case N > 1 is squarefree. This case is simle because there are only two ossibilities for the local comonents π, the Steinberg reresentation and its unramified quadratic twist, and π is determined by the Atin Lehner eigenvalue. The roof relies on a trace formula for roducts of Atin Lehner oerators on S (N due to Yamauchi [Yam73], which we translate to S new (N in Section 1. It was already nown that such dimensions can be comuted in rincile in this way, and some cases have been done before: the rime level case is in [Wa14], asymtotics for dimensions of lus and minus saces were given in [ILS00], and [HH95] gave a formula for the full cus sace in level 2 with rescribed Atin Lehner eigenvalues. So while the derivation is not esecially novel, we hoe the exlicit formulas and their consequences (articularly the biases discussed below may be of interest. In fact, our motiviation was different from [HH95], [ILS00] and [Wa14], which all had mutually distinct motivations. We emhasize that we are able to obtain quite simle formulas thans to the squarefree assumtion. The trace formula in [Yam73] is valid for arbitrary level, but Date: Aril 19,

2 becomes considerably more comlicated. In rincile, our aroach gives dimensions of saces with rescribed Atin Lehner eigenvalues in non-squarefree level also, but the resulting formulas may be messy. In any case, this would not give us dimensions for forms with secified local comonents π for non-squarefree levels. Our motivation in comuting these dimensions comes from two sources. First, this allows us to get very recise results about the distribution of signs of global functional equations and sign atterns for collections of Atin Lehner oerators. Various equidistribution results are nown about local comonents at unramified laces, or equivalently, Hece eigenvalues at rimes away from the level. For instance, and erhas most analogous, distributions of signs of unramified Hece eigenvalues are considered in [KLSW10]. At ramified laces, the most general results we now of for GL(2 are by Weinstein [Wei09], where he roves equidistribution of local inertia tyes for general level as the weight tends to infinity. However, these local inertia tyes do not distinguish unramified quadratic twists, and thus give no information in the case of squarefree level. While the fact that equidistribution holds is not at all surrising, the recise results we obtain about distribution and bias were erhas not exected. Let us exlain this in more detail. For the rest of the aer, assume N > 1 is squarefree and 2 is even. In Section 2 we obtain the dimension formulas for the lus and minus saces. This imlies the root number is equidistributed between +1 and 1 and the difference between the dimensions of the lus and minus saces is essentially indeendent of. It is also subolynomial in N recisely O(2 ω(n, where ω(n is the number of rime divisors of N. This is a considerable imrovement uon an earlier equidistribution result of Iwaniec Luo Sarna [ILS00, (2.73] which just imlies the difference is O((N 5/6. Moreover in any fixed sace S new (N, +1 always occurs at least as often as 1, i.e., there is a strict bias toward +1, and the size of this bias is on the order of the class number h Q( N. We initially found this bias surrising, but David Farmer exlained to us how such a bias (though erhas not the size is actually redicted by the exlicit formula. We briefly exlain this, and how the arithmetic of quaternion algebras also suggest (in fact rove for S 2 ( this bias. Moreover, we determine for which fixed saces S new (N there is erfect equidistribution (i.e., +1 occurs exactly as often as 1: for N = 2, 3 it merely deends on a congruence condition on, but for N > 3 it only haens twice, for S2 new (37 and S2 new (58. Next, fix M N with M > 1. In Section 3, we loo at distributions of sign atterns of Atin Lehner eigenvalues on S new (N for rimes M. Since the global root numbers are ( 1 /2 times the roduct of the local signs, this is a refinement of looing at distributions of root numbers. We obtain simultaneous equidistribution of sign atterns in both the weight and level (fixing M but varying N. As with the case of root numbers, the error term in the asymtotic is constant when varying the weight and O(2 ω(n when varying the level. We also find biases for certain sign atterns. On a fixed sace S new (N there is a otential bias which is toward or away from, deending on a arity condition collections of signs being 1 (i.e., local comonents being Steinberg. Here otential bias means there is a nonstrict inequality of dimensions when M < N. However, one gets a strict bias (a strict inequality when M = N, in which case the arity condition agrees with N the bias toward root number +1. Desite this otential bias, if M is divisible by rimes satisfying certain congruence conditions, the sign atterns for M are 2

3 erfectly equidistributed in fixed saces S new (N. One might thin of these two henomena as saying sign atterns are equidistributed with a bias in the weight but erfectly equidistributed in the level. Finally, in Section 4, we give an exlicit bound K such that for any > K all sign atterns occur in S new (N. This gives a lower bound on the number of Galois orbits in S new (N. Conjecturally, this lower bound equals the number of Galois orbits for sufficiently large (see [Tsa14]. Thus our bias of root numbers suggests that Galois orbits tend to be slightly larger for newforms with root number +1. Our second motivation for obtaining these dimension formulas was to aly them to the arithmetic of quaternion algebras to refine some results of [Mar] regarding Eisenstein congruences. We address this in the searate aer [Mar2]. Secifically, [Mar2] relates the distribution of Atin Lehner sign atterns to roerties of Sideal classes of quaternion algebras and to the existence of many congruences (both Eisenstein and non-eisenstein of modular forms mod 2. There, the formulas from the resent aer are used to give elementary criteria for the existence of congruences mod 2. We also suggest another ossible use of one of our formulas: the dimension formula for S new, (N tells us the dimension of the Saito Kuroawa sace of degree 2 Siegel modular forms of aramodular level N and weight 2 + 1, and thus may be useful when investigating aramodular forms of squarefree level. As one self-chec for correctness, we comared our formulas for small weights and levels with nown modular forms calculations via a combination of Sage and LMFDB. Acnowledgements. We are grateful to David Farmer, Abhishe Saha and Satoshi Waatsui for helful discussions and ointers to relevant literature. The author was artially suorted by a Simons Foundation Collaboration Grant. 1. Traces of Atin Lehner oerators In this section, we give an exlicit formula for the trace of a roduct of Atin Lehner oerators on the new sace of even weight 2 and squarefree level N. A formula on the full sace of cus forms S (N was originally given by Yamauchi [Yam73] for arbitrary level (also with unramified Hece oerators, though his final formula contains errors (e.g., the first term on the right hand side of the statement of Theorem 1.6 is missing the factor (nn 0 1 /2 coming from case (e of a(s on Sorua and Zagier rovide a corrected version in [SZ88]. First we will state the corrected version in the case of squarefree level. Throughout, M denotes a divisor of our squarefree N, M = N/M, and we assume M > 1. For N, let W denote the -th Atin Lehner oerator, and W M = M W. If W is an oerator on a vector sace S, we denote its trace by tr S W to clarify the vector sace. For < 0 a discrminant, let h( be the class number of an order O of discriminant, and w( = 1 2 O. Put h ( = h( if < 4 but h ( 4 = 1 2 and h ( 3 = 1 3. Define { x 1 y 1 (s = x y s ±2 1 s = ±2 3

4 where x, y are the roots of X 2 sx + 1. Put r(d, n = #{r mod 2n r 2 D mod 4n} and let δ i,j be the Kronecer delta function. Theorem 1.1 ([Yam73]; [SZ88]. For squarefree N, the trace tr S (N W M equals (1.1 1 (s/ M 2 s f h ( s2 4M f 2 t r( s2 4M f 2 (M /t 2, t + δ,2 where F = F s N is such that (s 2 4M/F 2 is a fundamental discriminant. Here s runs over integers such that s 2 < 4M and M s, f runs over ositive divisors of F which are rime to M, and t runs over ositive divisors of M such that M t F f. This formula is already considerably simler than in the case of non-squarefree level, and next we will exlicate it in a more elementary form. We will use the following elementary facts about r(d, n: r(d, n is multilicative in n, and if D is a discriminant then r(d, = 1 + ( D. First we consider the s = 0 terms. Note when M > 3, there is only an s = 0 term in (1.1. Assume s = 0. Then (s = ( 1 2 1, and F = 2 or F = 1 according to whether M is 3 mod 4 or not. Then t = M excet in the case that F = 2, f = 1 and M even which gives t { M 2, M }. If M 1, 2 mod 4, then the s = 0 summand in (1.1 is simly (1.2 ( h ( 4Mr( 4M, M. If M 3 mod 4, then the s = 0 summand is ( (1.3 ( h ( 4M(r( 4M, M + r( M, M 2 + h ( Mr( M, M, where we interret r( M, M 2 = 0 if M is odd. Suose M 3 mod 4. Then the well-nown relation between the class number of a maximal order and a non-maximal order tells us h ( 4M = (2 ( M 2 h ( M. Also note r( 4M, M = r( M, M odd, where M odd is the odd art of M. Further, r( M, M = r( M, 2r( M, M 2 = ( 1 + ( M 2 r( M, M 2 if M is even. We can ut all cases together as follows. Let a(m, M be defined as follows: a(m, M a(m, M M mod 8 for M odd for M even 1, 2, 5, Let M be the discriminant of Q( M. Then the s = 0 contribution to (1.1 is (1.4 ( a(m, M h ( M r( M, M odd. 4

5 The only other terms in (1.1 are the terms s = ±M when M = 2, 3. comute these now. We first calculate 1 0 mod 8 (± 1 2 mod 8 2 = 1 4 mod mod 8 and 1 0 mod mod 12 (± 2 4 mod 12 3 = 1 6 mod mod mod 12. In these cases s 2 4M is 4 or 3 as M is 2 or 3, so F = f = 1 and t = M. Thus each s = ±M summand of (1.1 is (1.5 ( M 1 M r(6 M, M. Hence in all cases we may rewrite (1.1 as (1.6 tr S (N W M = 1 2 ( 1 2 a(m, M h ( M r( M, M odd 1 2 δ M,2 ( 2r( 4, M 1 3 δ M,3 ( 3r( 3, M + δ,2. To get the trace on the new sace, we will use the following formula. For n N, let ω(n be the number of rime divisors of n. Proosition 1.2 ([Yam73]. For N squarefree, M N and M = N M, we have tr S new (N W M = d M ( 2 ω(m /d tr S (dm W M, We will comute this weighted sum over d M of each term in (1.6. First note the binomial theorem imlies (1.7 /d = ( 1 ω(m. d M ( 2 ω(m Lemma 1.3. Suose M is odd and D is a discriminant rime to M. Then ( 2 ω(m /d r(d, d = (( { ( D ( 2 ω(m D = 1 for all M 1 = 0 else. d M M Proof. Let M 1 be the roduct of M such that ( D = 1. Then the above sum is ( 2 ω(m /d ( ( D 1 + = ( 2 ω(m /M 1 ( 2 ω(m 1 /d 2 ω(d. d M d d M 1 The sum on the right is ( ω(m 1 = 0 if M 1 > 1 and 1 if M 1 = 1. (Alternatively, one can realize the sum as a Dirichlet convolution. 5 We

6 Noting that the only deendence of a(m, M on M is on the arity of M, this lemma combined with the above roosition is enough to get an exlicit formula for tr S new (N W M when M is odd. So let us consider the case that M is even. To aly the revious lemma to this case, we note that ( 2 ω(m /d f(d = ( 2 ( 2 ω(m odd /d f(d + ( 2 ω(m odd /d f(2d, d M d M odd d M odd for a function f on N. Combining this with (1.7, the lemma, and the facts that r( 4, 2 = 1 and r( 3, 2 = 0, when M is even we see tr S new (N W M is 1 2 ( 1 (( 2 h M ( M ( 2a(M, 1 + a(m, 2 1 M odd δ M,2 ( 2 M odd (( δ M,3 ( 3 M odd (( ( 1 ω(m δ,2. We now summarize the formula for both cases, M odd and M even, together. Let b(m, M = a(m, 1 when M is odd and b(m, M = 2a(M, 1 + a(m, 2 when M is even, i.e., b(m, M is given as follows: b(m, M b(m, M M mod 8 for M odd for M even 1, 2, 5, Proosition 1.4. Let N be squarefree, M N with M > 1, M = N/M and M the discriminant of Q( M. Then tr S new (N W M = 1 2 ( 1 (( 2 h ( M b(m, M M 1 + δ,2 ( 1 ω(m δ M,2 ( 2 2 M odd M (( 4 1 δ M,3 ( 3 3 M The following bound will be useful for equidistribution results. Corollary 1.5. With notation as in the roosition, we have (( 3 1. tr S new (N W M 2 ω(m odd +1 h( M + δ,2. The next result will give us finer information for erfect equidistribution. Corollary 1.6. With notation as in the roosition, suose M > 3. If 4, then tr S new (N W M = 0 if and only if one of the following holds: (i ( M = 1 for some odd M, or (ii M is even and M 7 mod 8. If = 2 and dim S 2 (N > 0, then tr S new 2 (N W M = 0 if and only if N = M with M {37, 58} or N = 2M with M {13, 19, 37, 43, 67, 163}. 6

7 Proof. The 4 case is evident. For = 2, trace 0 can only haen if exactly one of h( M, b(m, M and ω(m odd + 1 is 2, and the other two are 1. There is no M > 2 with M 3 mod 4 such that h( M = 1, so we cannot have ω(m odd = 1, and thus N {M, 2M}. If b(m, M = 2 so M 7 mod 8 and N = M, then h( M = 1 imlies M = 7 but dim S 2 (7 = 0. If b(m, M = 2, so M 3 mod 8 and M = 2N, then h( M = 1 imlies M {11, 19, 43, 67, 163}, but dim S 2 (2M = 0 for M = 11. The other 4 ossibilities for M all give trace 0 on nonzero saces. Lastly, if b(m, M = 1 so M 3 mod 4 and h( M = 2, then M is one of 5, 6, 10, 13, 22, 37 and 58. One only gets nonzero newsaces when N = M and M = 37, 58 or N = 2M and M = 13, Refined dimension formulas I: lus and minus saces As a first alication, we consider the distribution of signs of function equations. If f S (N is a newform, then the sign of the functional equation w f of the L- series L(s, f is ( 1 /2 times the eigenvalue of W N. Let S new,± (N be the subsace of S (N generated by newforms f with w f = ±1. For convenience, we recall the exlicit formula for the full new sace given by G. Martin. Theorem 2.1 ([Mar05]. For N squarefree, ( dim S new ( 1ϕ(N 1 (N = (( N ( (( 1 + δ,2 µ(n. 3 3 Note (2.1 dim S new,± (N = 1 2 ( N dim S new (N ± tr S new (N W N. This combined with Proosition 1.4 gives the following exlicit formulas for dim S new,± (N. Theorem 2.2. Suose N > 1 is squarefree. When N > 3, dim S new,± (N = 1 dim Snew (N ± ( 1 2 h( Nb(N, 1 δ,2 where we recall N is the discriminant of Q( N and b(n, 1 = 1, 2 or 4 according to whether N 3 mod 4, N 7 mod 8 or N 3 mod 8. When N = 2 and > 2, dim S new,± (2 = 1 dim Snew (2 + 2 When N = 3 and > 2, dim S new,± (3 = 1 dim Snew (3 + 2 { ± 1 2 0, 2 mod 8 0 else., { ± 1 2 0, 2, 6, 8 mod 12 0 else. We note that the N > 3 rime case of this result is essentially contained in [Wa14] (also using [Yam73], though the minus sign in Theorem 3.2 of [Wa14] should be ignored. 7

8 Corollary 2.3. Fix N > 1 squarefree. dim S new, (N, and in fact For any, we have dim S new,+ (N dim S new,+ (N dim S new, (N = c N h( N δ,2, where c N { 1 2, 1, 2} is indeendent of. In articular, as along even integers we have dim S new,± ( 1ϕ(N (N = + O(1. 24 The corollary says that the global root numbers w f for fixed (squarefree level are equidistributed as the weight goes to infinity, but are biased toward +1 for bounded weights. Since the difference between dim S new,+ (N and dim S new, (N is a simle multile of h( N for 4, the bias roughly grows lie N in the (squarefree level (though as a roortion of the total dimension, goes to zero as. We can also get an asymtotic for varying the level relacing the O(1 in the last statement by O(2 ω(n (cf. Corollary 3.4. This is a significant imrovement on the asymtotic [ILS00, (2.73]. Next we consider for what saces there is erfect or near erfect distribution of the root numbers. For N = 2, 3, the behavior is clear from the theorem. Namely, we always have dim S new,+ (N dim S new, (N {0, 1}, with dim S new,+ (N = dim S new, (N if and only if 4, 6 mod 8 when N = 2, and if and only if 4, 10 mod 12 when N = 3. Corollary 2.4. Suose N > 3 is squarefree. Then dim S new,+ (N = dim S new, (N if and only if dim S new (N = 0 or = 2 and N {37, 58}. When 4, we have that dim S new,+ (N = dim S new, (N + 1 if and only if N {5, 6, 7, 10, 13, 22, 37, 58}. In general, for fixed and r Z 0, there are only finitely many squarefree N such that dim S new,+ (N = dim S new, (N + r. Proof. The first assertion follows from (2.1 and Corollary 1.6, and the second follows from the = 2 comutations in the roof of said corollary. Hence erfect equidistribution of root numbers is very rare, which will be in contrast to the situation for incomlete sign atterns in the next section. Here are a coule of heuristic reasons for the bias towards root number +1 and thus the rareness of erfect equidistribution. One reason comes from the hilosohy that L-functions which barely exist tend to have negative signs, indly exlained to us by David Farmer. This is formulated in [FK] where the focus is on the second Fourier coefficient, but the same reasoning there alies to signs of functional equations. Namely, if f S new (N, the exlicit formula for L(s, f exresses the sum of ϕ(ρ in terms of log N and some other terms, where ϕ is a suitable test function and ρ runs over nontrivial zeroes of f. Taing ϕ to be a test function as in [FK, Thm 3.2] essentially forces N to be larger when L(s, f has a zero at the central oint, in articular when the root number is 1. Thus one might exect root number +1 more often, at least for small levels. Here is another reason coming from the arithmetic of quaternion algebras. Suose N = is rime and = 2. Let B/Q the definite quaternion algebra of discriminant. Then the class number h B = 1 + dim S2 new ( and the tye number 8

9 t B = 1 + dim S new,+ 2 (. Since 1 2 h B t B h B, we see that dim S new,+ 2 ( can vary between 1 2 dim Snew 2 ( 1 2 and dim Snew 2 (. This suggest a bias toward root number +1, and one could comare tye and class number formulas to get another roof of Theorem 2.2 when N = and = 2. We discuss this connection between dimension formulas and arithmetic of quaternion algebras further in [Mar2]. 3. Refined dimension formulas II: local sign atterns In this section, we obtain an exlicit formula for the sace of newforms with fixed Atin Lehner eigenvalues at rimes dividing M, and obtain consequences for the distribution of this collection of eigenvalues. By a sign attern ε M for M, we mean a multilicative function d ε M (d on the divisors d of M such that ε M (1 = 1 and ε M ( {±1} for each M. Define S new,ε M (N to be the subset of S new (N generated by newforms f such that the Atin Lehner eigenvalue of f is ε M ( for each M. Lemma 3.1. Fix two sign atterns ε, ε for a squarefree M. Then ε(dε (d = d M { 2 ω(m ε = ε 0 else. Proof. Let S = { M : ε( ε (}. The ε = ε case is obvious, so assume S. Note ε(dε (d = 1 if and only if the number of S such that d is even. Now recisely half the divisors d of M satisfy this roerty because exactly half the divisors of S have an odd number of rime factors. (If ε(m = 1 or ε (M = 1, one can also realize this sum as a Dirichlet convolution. Proosition 3.2. Let N be squarefree, 1 < M N, and ε M a sign attern for M. Then dim S new,ε M (N = 2 ω(m ε M (d tr S new (N W d, d M where W 1 means the identity oerator. Proof. Consider the sum on the right. Note for any sign attern ε M of M, each term on the right gives a contribution of ± dim S new,ε M (N. The sign in the contribution is recisely ε M (dε M (d. Hence by the above lemma, the sum aearing on the right is just 2 ω(m dim S new,ε M (N. This roosition combined with Proosition 1.4 gives an exlicit formula for dim S new,ε M (N. For simlicity, we just state it when there are no extra M = 2 or M = 3 terms arising from Proosition 1.4. Theorem 3.3. Let N be squarefree, M N, and ε M a sign attern for M. Assume for simlicity that (i 2 M or ( 4 = 1 for some N M, and (ii 3 M or ( 3 = 1 for some odd N M. Let S be the set of divisors d > 1 of M such that ( d = 1 9

10 for all odd N d. Put ω (n = ω(n odd. Then dim S new,ε M (N = 1 ( dim S new 2 ω(m (N ( 1 2 ε M (dh ( d b(d, N/d( 2 ω (N/d d S + δ,2 ( 1 ω(n( (1 ε M ( 1 The roosition and the theorem yield results about equidistribution of sign atterns. Corollary 3.4. Let M > 1 be squarefree and ε M a sign attern for M. As N where is an even integer and N is a squarefree multile of M, M dim S new,ε ( 1ϕ(N M (N = 2 ω(m 12 + O(2ω(N. Proof. Note that 2 ω(m times the right hand side is an asymtotic for the full new sace. Now use Proosition 3.2 and note that naively summing the bounds in Corollary 1.5 gives an error bound which is O(2 ω(n. Note this gives simultaneous equidistribution of sign atterns for fixed M in both weight and level, where the error term is O(1 if we fix (or just bound the level. The error term can be made recise if desired. We remar this imlies the equidistribution art of Corollary 2.3 by taing N = M and summing over sign atterns with ε M (M + 1 or 1, but the exlicit error term obtained in this way will be worse. Now if we fix the level N, we might as if there is any bias in the collection of all ossible sign atterns, similar to the bias we saw for global root numbers. Note if N is rime, then the sign atterns simly corresond to the global root numbers. So let us begin by considering the case N = q for distinct rimes, q > 3 and tae 4. Then the relevant art of the formula in Theorem 3.3 for a sign attern ε for N = M is (3.1 ( 1 2 ( 2(δ S ε(h( b(, 1 + δ q S ε(qh( q b(q, 1 + ε(qh( q b(q, 1, where δ d S is 1 if d S and 0 otherwise. Secifically, there will be a bias towards (res. away from ε, i.e., dim S new,ε (N is greater (res. less than 2 ω(n dim S new (N if and only if (3.1 is ositive (res. negative. For simlicity, assume 0 mod 4, so the biases we describe will be flied if 2 mod 4. We write the ossible sign atterns ε for N = q as ++, +, and so on, where the first sign is the sign of ε( and the second is the sign of ε(q. If, q S, then only the last term of (3.1 aears, and hence we get an equal bias toward each of ++ and and the same bias away from each of + and +. If S but q S, then (3.1 will be ositive and maximized when ε( = ε(q = 1, so there is a definite bias towards, and similarly a bias away from +. For the signs ++ and +, the actual values of class numbers and b(, 1 come into lay, and it is not clear which has a ositive or negative bias, but at least we can say the bias will not be as strong for and +. The case of S, q S is similar so we lastly suose both, q S. As in the revious case, there is a clear and maximal bias toward the sign attern, 10

11 though the bias to or away from any of the other sign atterns deends on class numbers and congruences of divisors. So while there does not aear to be any nice uniform descrition of the biases for general sign atterns, there is at least a clear bias for in all cases. The same argument generalizes to the following. Corollary 3.5. Let 4, M, N squarefree odd with M N and M > 3. If M is divisible by 3, assume there is an odd N M such that ( 3 = 1. Let M be the sign attern for M given by M ( = 1 for each M. Then for any other sign attern ε for M, we have dim S new, M (N dim S new,ε (N if + ω(n is even, 2 dim S new, M (N dim S new,ε (N if + ω(n is odd. 2 Moreover, if M = N the above inequalities are strict for at least one choice of ε. Proof. The above assumtions guarantee that each term in the sum over d S in Theorem 3.3 has sign ( 1 ω(n. If M = N, we always have N S so this sum is nonzero. We note this bias to or away from N agrees with the bias toward global root number +1. For examle, suose M = N = 35. In weight 4 there are 3 Galois orbits, one of size 1 with signs +, one of size 2 with signs ++, and one of size 3 with signs (root number +1. In weight 6, there are 4 Galois orbits, one for each ossible sign attern, with sizes 1, 2, 3, 4 and the smallest orbit has signs (root number 1. When M = N is a roduct of an odd number of factors with M, N as in the corollary, the same argument also gives a bias toward N when = 2. On the other hand, the theorem also gives us erfect equidistribution of sign atterns when moving to to sufficiently large level. Corollary 3.6. Fix an even 4. Let M > 1 be squarefree and ε M, ε M be any two sign atterns for M. Let S be a set of odd rimes M such that for any d M, ( d = 1 for some S. (If M is odd, we can omit the d = 1 case of this condition. Then for any squarefree N divisible by both M and each S, we have dim S new,ε M (N = dim S new,ε M (N. For instance, let M = 10 and S = {3, 13}. Note ( ( 4 13 = 40 ( 13 = 1 and 8 ( 3 = 20 3 = 1. Hence for any squarefree N which is a multile of 390, all sign atterns occur equally often for the Atin Lehner eigenvalues at 2 and 5 among the newforms of level N and a fixed weight 4. Put another way, the corollary says that if the rimes dividing N M satisfy certain congruence conditions, then we have erfect equidistribution of sign atterns for M. So if we thin about starting with a fixed M and, and successively raising the level by randomly adding other rime factors, then with robability 1 we will eventually reach a state of erfect equidistribution. Thus we may thin of this as saying there is erfect equidistribution in the level. Note that even though Corollary 3.5 gives a bias to or away from M, each time we add a rime to the level in this rocess, we fli the sign of this bias, so even before we hit a fixed level 11

12 N with erfect equidistribution, variation in the distribution of the sign atterns aears to oscillate. 4. Bounds on number of Galois orbits Let f S new (N be a newform. By the sign attern for f, we mean a sign attern ε for N such that ε( is the eigenvalue of the -th Atin Lehner oerator for f, for all N. We say ε occurs in S new (N if it is the sign attern of some newform f S new (N. More generally, if ε M is a sign attern for M N, we say it occurs in S new (N if S new,ε M (N 0. Clearly, if two new forms f, g S new (N have different sign atterns, then f, g lie in different Galois orbits. Thus the number of sign atterns occurring in S new (N rovides a lower bound on the number of Galois orbits. A generalization of Maeda s conjecture asserts that for squarefree level and sufficiently large weight, the number of Galois orbits is recisely the number of ossible sign atterns, 2 ω(n ([Tsa14]; see also [CG15]. One consequence of our results gives an effective bound on weights with at least this many Galois orbits. However, we remar that in some low weights the number of Galois orbits is strictly larger than the number of sign atterns which occur e.g., there are 2 Galois orbits in S6 new (17 with Atin Lehner eigenvalue +1 at 17. Hence, even admitting the truth of this generalized Maeda conjecture, one may need to tae still higher weights to get exactly 2 ω(n orbits. We also note that the generalized Maeda conjecture together with our results on bias of signs would imly that there is a bias in the distribution of the size of Galois orbits when searating by root number or sign atterns. In articular, the average size of Galois orbits of newforms with root number +1 should be larger than that for newforms with root number 1 for fixed N,, though these averages should be asymtotic as N. In fact, looing at tables in LMFDB for small N suggests there may be a tendency for Galois orbits with root number +1 to be larger on average already in weight = 2. Proosition 4.1. Fix M N squarefree with M > 1. Let H M = max{h( d : d M} and K N,M = 24(3ω(N 2 ω(nodd H M ω(n + 1. ϕ(n Then for any even > min{k N,M, 3}, all ossible sign atterns for M occur for S new (N, and thus there are at least 2 ω(m Galois orbits in S new (N. Proof. We just need to show that the d = 1 term in Proosition 3.2 for M = N is larger in absolute value than the sum of all the other terms with d N. Now comare Theorem 2.1 and Corollary 1.5. Here we majorized the sum d M (N/d 2ω with the same sum taen over d N. We did not strive for otimality with this bound. For instance, one can imrove the bound for N along various families by woring with a set S as in the statement of Theorem 3.3. When N = M, this gives a lower bound on the weight to get all ossible sign atterns and as N, note that K N,N grows at a slower rate than H N, which grows roughly at a rate of N. On the other hand, for fixed M and N, K N,M 1, giving all sign atterns for a fixed weight and large level. (A simle 12

13 modification of the bound will treat = 2. This agrees with the equidistribution results on sign atterns in Corollary 3.4 for a fixed weight and varying level. In the simle case of rime level, we get the following imroved exlicit bounds. Proosition 4.2. For 13 both sign atterns for occur in S new ( for all 4. When {7, 11} (res. = 5, the same is true for 6 (res. 8. When = 2, both sign atterns for occur in S new ( if and only if > 60 and 71, or = 37. Proof. From Theorem 2.2, both sign atterns occur in S new ( whenever > 6b(, 1h( , 1 for 4 and > 3. Now the class number formula combined with an exlicit bound on Dirichlet L-values (see [Coh07, Pro ] tells us h( π ( 1 2 log + log log Since b(, 1 4, the above bound on using this estimate is less than 4 for > 157 and less than 2 for > Using exact class number calculations (and b(, 1 rather than 4, we see the above bound on is less than 4 for all > 11, and less than 2 for all > 60 excet {71, 79, 83, 89, 101, 131}. Exlicit calculations finish the = 2 case. For = 5, 7, 11 this bound resectively gives the result for 10, 8, 6. Exlicit calculation of these saces then shows the stated bounds on hold (and are otimal for {5, 7, 11}. Note for = 2, 3, Theorem 2.2 imlies both signs occur in S new ( whenever dim S new ( 2. When = 2, this haens for {14, 20, 22} or 26. When = 3, this haens for = 10 or 14. References [CG15] Sam Chow and Alexandru Ghitza, Distinguishing newforms, Int. J. Number Theory 11 (2015, no. 3, , DOI /S MR [Coh07] Henri Cohen, Number theory. Vol. II. Analytic and modern tools, Graduate Texts in Mathematics, vol. 240, Sringer, New Yor, MR [FK] David Farmer and Sally Koutsoliotas, The second Dirichlet coefficient starts out negative (2015, rerint. arxiv: v1. [HH95] Yuji Hasegawa and Ki-ichiro Hashimoto, On tye numbers of slit orders of definite quaternion algebras, Manuscrita Math. 88 (1995, no. 4, [ILS00] Henry Iwaniec, Wenzhi Luo, and Peter Sarna, Low lying zeros of families of L- functions, Inst. Hautes Études Sci. Publ. Math. 91 (2000, (2001. [KLSW10] E. Kowalsi, Y.-K. Lau, K. Soundararajan, and J. Wu, On modular signs, Math. Proc. Cambridge Philos. Soc. 149 (2010, no. 3, [Mar05] Greg Martin, Dimensions of the saces of cus forms and newforms on Γ 0 (N and Γ 1 (N, J. Number Theory 112 (2005, no. 2, [Mar] Kimball Martin, The Jacquet-Langlands corresondence, Eisenstein congruences, and integral L-values in weight 2 (2016, Math. Res. Let., to aear. [Mar2], Congruences for modular forms mod 2 and quaternionic S-ideal classes (2016, Prerint, arxiv: [SZ88] Nils-Peter Sorua and Don Zagier, Jacobi forms and a certain sace of modular forms, Invent. Math. 94 (1988, no. 1, [Tsa14] Panagiotis Tsanias, A ossible generalization of Maeda s conjecture, Comutations with modular forms, Contrib. Math. Comut. Sci., vol. 6, Sringer, Cham, 2014, [Wa14] Satoshi Waatsui, Congruences modulo 2 for dimensions of saces of cus forms, J. Number Theory 140 (2014, [Wei09] Jared Weinstein, Hilbert modular forms with rescribed ramification, Int. Math. Res. Not. IMRN 8 (2009,

14 [Yam73] Masatoshi Yamauchi, On the traces of Hece oerators for a normalizer of Γ 0 (N, J. Math. Kyoto Univ. 13 (1973, Deartment of Mathematics, University of Olahoma, Norman, OK