COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL

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1 COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL JEFFERY BREEDING II, CRIS POOR, AND DAVID S. YUEN Abstract. This article gives upper bounds on the number of Fourier-Jacobi coefficients that determine a paramodular cusp form in degree two. The level N of the paramodular group is completely general throughout. Additionally, spaces of Jacobi cusp forms are spanned by using the theory of theta blocks due to Gritsenko, Skoruppa and Zagier. We combine these two techniques to rigorously compute spaces of paramodular cusp forms and to verify the Paramodular Conjecture of Brumer and Kramer in many cases of low level. The proofs rely on a detailed description of the zero dimensional cusps for the subgroup of integral elements in each paramodular group. Contents 1. Introduction 1 2. Siegel modular forms and notation 5 3. Jacobi forms 6 4. Theta blocks 8 5. Formal Fourier-Jacobi expansions Vanishing theorems Determining numbers of Fourier-Jacobi coefficients Coset representatives of Γ 0N\Sp 2 Z/P 2,0 Z Appendix 38 References Introduction A finite dimensional vector space of Siegel modular forms, M k Γ, is determined by a finite set of Fourier coefficients. Specifying such determining sets without necessarily knowing the dimension of the space is a problem that dates back to Siegel. A related problem asks for the number of Fourier-Jacobi coefficients that determines M k Γ. Eichler showed that the Fourier-Jacobi coefficients up to index 2 3 µ 2 n k 4π determine the space M k Sp n Z; here, µ n is Hermite s constant. Generalizations and improvements were given in [24] and [27] but were still restricted to level one. Such results provide one avenue to rigorous computations. The Paramodular Conjecture [4] has recently focused interest on the paramodular groups KN in degree two. Date: August 22, Mathematics Subject Classification. 11F46, 11F50. Key words and phrases. paramodular, theta block, Fourier-Jacobi. 1

2 2 JEFFERY BREEDING II, CRIS POOR, AND DAVID S. YUEN Theorem 1.2 below presents an upper bound on the number of Fourier-Jacobi coefficients that determine S k KN, χ, the vector space of weight k paramodular cusp forms of level N that are eigenforms under all the paramodular Atkin-Lehner involutions with signs specified by a character χ trivial on KN. Determining sets of Fourier coefficients are also given. The case of prime level in Theorem 1.2 was first proven in [21]. Our application of Theorem 1.2 is an appealing strategy for computing individual spaces of paramodular cusp forms. The Fourier-Jacobi coefficients of paramodular forms are Jacobi forms and so Theorem 1.2 allows us to control spaces of paramodular forms by spanning a finite set of spaces of Jacobi forms, which is more tractable. These rigorous computations for low kn, are given in Tables 1, 2 and 3. Weight 2 paramodular cusp forms occur in the Paramodular Conjecture and, for composite levels, our dimension results are new. Nontrivial weight 3 paramodular cusp forms provide canonical divisors on the moduli space of 1, N polarized abelian surfaces. When the moduli space is rational or unirational, it follows that S 3 KN = {0}. Gritsenko [8] proved that S 3 KN {0} for N > 36, so our Table 2 completely enumerates the twenty cases of dimension zero. For sixteen of these cases the rationality or unirationality of the moduli space is known, compare the work of Gross and Popescu [13]; our vanishing results for the four cases N = 15, 24, 30, 36 are new. For composite N > 4, all the nonzero dimensions are new. For prime level p, the dimensions of S k Kp for k 3 are known by work of Ibukiyama [17][19]; Ibukiyama and Kitayama plan to publish the dimension of S k KN for squarefree N and k 3, see [20]. The statement of Theorem 1.2 uses the Jacobsthal function and it is interesting to see this function arise naturally in automorphic form theory. The proof of Theorem 1.2 relies on a detailed description of the double cosets KN Sp 2 Z \Sp 2 Z/P 2,0 Z for general N and this description is given separately for the case of prime powers in Theorem 1.3. The coset description for prime levels was already known by the work of Ibukiyama and Hashimoto [16]. Definition 1.1. The Jacobsthal function jn is defined to be the smallest positive integer m such that every sequence of m consecutive positive integers contains an integer coprime to N. Theorem 1.2. Let k, N N. Let χ : KN {±1} be a character trivial on KN. Let f S k KN, χ be a common eigenfunction of the paramodular Atkin-Lehner involutions and have Fourier-Jacobi expansion p r N f = φ jn ξ jn. j=1 Let N = p α1 1 pα l l be the prime factorization of N and set Ñ = p 1 p l. Choose µ N such that 2µ + 1 jñ/p i for all i. Let κ be 1 when N is prime, 2 when N is a composite prime power and 1 + µ + µ 2 otherwise. If φ jn = 0 for j κ k 10 N p r + p r 2 p r, then f = 0. When N is a composite prime power the + 1 inequality may be taken strictly.

3 COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL 3 We state the double coset decomposition for the important case of prime powers q r in Theorem 1.3. For q N and x, y, z, M Z, define the following matrix: W q; x, y, M, z = Mq x 1 0. Mq x zm 2 q y 0 1 Theorem 1.3. Let q N be prime. Let r Z be nonnegative. We have the disjoint double coset decomposition Sp 2 Z = Kq r Sp 2 Z W q; r µ ν, r ν, M, zp 2,0 Z, µ,ν, M, z where the indices range over 0 µ r/2, M Z/q µ Z /{±1}, 0 ν r 2µ, and z Z/q ν Z /squares for 2µ + 2ν r and z Z/q r 2µ ν Z /squares for 2µ + 2ν > r. We now explain how Theorem 1.2 and its companions in section 7 make rigorous a strategy for computing spaces of paramodular forms that had hitherto been heuristic. Denote by S k KN ɛ the subspace of S k KN with eigenvalue ɛ = ±1 under the Fricke involution. For a paramodular cusp form f S k KN ɛ, let f τ z ω z = m=1 φ Nmτ, ze Nmω be the Fourier-Jacobi expansion, so that the φ Nm are Jacobi cusp forms of weight k and index Nm. Following a method of Aoki [1] for N = 1, it was pointed out in [21] that the involution condition 1 n, m, r Z, cn, r; φ Nm = ɛ cm, r; φ Nn is very strong. There it was asked whether any sequence φ Nm of Jacobi forms that satisfies 1 has a convergent series m=1 φ Nmτ, ze Nmω and is thus the Fourier-Jacobi expansion of some paramodular form. The cases of N = 2, 3, 4 were answered affirmatively in [21]. An improved upper bound on the dimensions of Atkin-Lehner subspaces S k KN, χ is obtained by combining the involution condition 1, and similar conditions for the other paramodular Atkin-Lehner involutions, with a determining set of Fourier-Jacobi coefficients. If the upper bounds are in fact the correct dimensions then S k KN ɛ can be rigorously computed by the construction of enough cusp forms. Of course, any algorithm, however foolish, that computes upper bounds can make the same claim. The real point is that, in the examples where we can say for sure, our improved upper bound is in fact the correct dimension. The examples in this article are thus further evidence that involution conditions alone imply convergence. Theoretical work has yet to explain the success of this strategy; for interesting work on similar topics see [33] and [5]. The method for computing upper bounds of dim S k KN ɛ given here is appealing because it works for general levels N and also for weight 2, a weight inaccessible to trace formulas in degree two. In order to span spaces of Jacobi cusp forms J cusp k,n, we used the theory of theta blocks introduced by Gritsenko, Skoruppa and Zagier in [14]. Spaces of Jacobi cusp forms are not always spanned by cuspidal theta blocks but, following a suggestion of D. Zagier, we have had in our examples complete success spanning spaces of Jacobi forms by linear combinations of weak Jacobi forms that are theta blocks. Our Table 1 of dim S 2 KN ɛ gives evidence for the truth of the Paramodular Conjecture of Brumer and Kramer [4]. Abelian surfaces A defined over Q of conductor N with End Q A = Z should correspond to Hecke eigenforms f S 2 KN ɛ

4 4 JEFFERY BREEDING II, CRIS POOR, AND DAVID S. YUEN that are not Gritsenko lifts and that have rational eigenvalues. In this correspondence, we have Ls, f, spin = Ls, A, Hasse-Weil. Furthermore, we should have ɛ = 1 when the rank of A is even and ɛ = 1 when the rank of A is odd. The smallest prime level for which S 2 Kp has a nonlift is p = 277, see [30]; the smallest known level for which S 2 KN has a nonlift is N = 249 = 3 83, see [32]. There are indeed abelian surfaces A/Q possessing these conductors and none for odd N < 249, see [4]. The rigorous computations given here provide more evidence for the Paramodular Conjecture in some cases N 60 by showing that the low levels N with no abelian surfaces over Q do not have rational paramodular nonlift eigenforms. Table 1 gives dim S 2 KN for N 60. In these tables, an omitted level N indicates that the dimension is zero. Each table also gives the best a priori upper bound from section 7 on the needed number of Fourier-Jacobi coefficients. These weight two spaces were all spanned by Gritsenko lifts, which is consistent with the Paramodular Conjecture. Table 1. Dimension of S 2 KN and number of FJ-coefficients needed in proof. N dim FJCs Table 2 gives the dimension of S 3 KN for N 40. These spaces were all spanned by Gritsenko lifts. All twenty levels with dimension zero are omitted. Table 2. Dimension of S 3 KN and number of FJ-coefficients needed in proof. N dim FJCs N dim FJCs Table 3 gives dim S 4 KN for N 40 and the dimension of nonlifts. Table 3. Dimension of lifts and nonlifts for S 4 KN. N dim FJCs

5 COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL 5 The signs in the tables label paramodular Atkin-Lehner spaces, see section 2 for the definition of the Atkin-Lehner involutions. Each row of signs is ordered by the distinct prime divisors of N and each sign in the row is the value of the Atkin- Lehner involution corresponding to the largest power of that prime dividing N. For example, dim{f S 4 K34 : f AL 2 = f and f AL 17 = f} = 2. In every case covered by these tables, the product of the signs is 1 k, so the signs need not be listed when N is a power of a single prime. N dim FJCs N dim FJCs nonlift Siegel modular forms and notation We set J = 0 I n. The general symplectic group of degree n over a ring R is I n 0 GSp n R := {g GL 2n R : ν GL 1 R : g Jg = νj}. The subgroup with ν = 1 is Sp n R. We refer to the textbook [15] for the general theory of Siegel modular forms. Let Γ be a discrete subgroup of Sp n R commensurable with Γ n = Sp n Z. Let H n be the Siegel upper half space and j : Sp n R H n C be the factor of automorphy given by jg, Z = detcz+d for g = C A D B. We write M kγ, χ for the C-vector space of Siegel modular forms of weight k and character χ with respect to Γ; that is, the holomorphic functions on H n, bounded at the cusps, that transform by the factor of automorphy j k χ. The subspace of cusp forms is denoted by S k Γ, χ. In the case of degree one, we consider Γ 0 N = N Sp 1 Z, for Z, and the extensions ˆΓ 0 N = Γ 0 N, I 2 and Γ + 0 N = Γ 0N, F N, where F N = N N 0 is the Fricke involution. Additionally we consider the group Γ 0 N obtained by adjoining all the Atkin-Lehner involutions. For α N, meaning α N and α, N α = 1, fix any x N 1, x 2, t 1, t 2 Z such that x 1 x 2 α t 1 t 2 α = 1 and set AL α = 1 x1α t 2 α t 1N x 2α and define Γ 0 N = Γ 0 N, {AL α } α N. For any N N, define the paramodular group in degree two to be N /N KN = { : Z} Sp 2 Q. N N N N

6 6 JEFFERY BREEDING II, CRIS POOR, AND DAVID S. YUEN The group KN has a normalizer given by µ N = F N 0 0 F N and we also call µ N the Fricke involution. If we adjoin µ N, we get the group KN + = KN, µ N. For α N, µ α = AL α 0 0 AL α is also a normalizer of KN with µ 2 α KN, see [10]. We let KN = KN, {µ α } α N denote the extension of the paramodular group by all these paramodular Atkin-Lehner involutions. A character χ : KN {±1} is called an Atkin-Lehner character if χ is trivial on KN. These characters form a group of order 2 t, where t is the number of distinct prime divisors of N. Throughout this article, when we write M k KN, χ, the weight k is in Z 0, the level N is in N and the character χ is an Atkin-Lehner character. We define, following [18], the standard groups Γ 0N = KN Sp 2 Z and 0 P 1,0 R = { 0 : R} SL 2 R, P 2,1 R = { 0 : R} Sp 2 R, P 2,0 R = { : R} Sp 2 R Define homomorphisms i 1, i 2 : SL 2 R Sp 2 R by i a b a 0 b 0 1 c d = c 0 d 0, i a b 2 c d = a 0 b c 0 d and homomorphisms u : GL 2 R Sp 2 R and t : M sym 2 2 R Sp 2R by ua = A 0 0 A, tb = I B 0 I. For n N, define φn = Z/nZ, ψn = {x Z/nZ : x 2 = 1}, { 1 if n 2, ϖn = φn/nψn, ξn = 2 if n Jacobi forms We define Jacobi forms following [9], the standard reference is [7]. Consider two types of elements in Γ 2, h = u λ t v 0 κ v and i a b 1 c d for λ, v, κ Z, and for a b c d SL2 Z. The subgroup of Γ 2 generated by the h is called the Heisenberg group HZ. The character v H : HZ {±1} is defined by v H h = 1 λv+λ+v+κ. The second type gives a copy of SL 2 Z inside Γ 2. This copy of SL 2 Z along with HZ and ±I 4 generate P 2,1 Z. The character v H extends uniquely to a character on P 2,1 Z that is trivial on the copy of SL 2 Z. Likewise, the factor of automorphy ɛ cτ + d of the Dedekind eta function extends uniquely to a factor of automorphy on P 2,1 Z H 1 C that is trivial on HZ and defines the multiplier ɛ : P 2,1 Z e 1 24 Z, where ex = e 2πix. We select holomorphic branches of roots that are positive on the purely imaginary elements of H n. For m Q, a, b, 2k Z, consider holomorphic φ : H 1 C C whose modified function φ : H 2 C, given by φ τ z ω z = φτ, zemω, transforms by the factor of automorphy j k ɛ a vh b for P 2,1Z. We necessarily have 2k a b mod 2 and m 0 for nontrivial φ. Such φ have Fourier expansions φτ, z = n,r Q cn, r; φqn ζ r, for q = eτ and ζ = ez. The support of such φ is suppφ = {n, r Q 2 : cn, r; φ 0}. If the support of φ has n bounded from below, we say φ is weakly holomorphic and write φ Jk,m wh ɛa vh b. Sometimes nearly holomorphic is used,

7 COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL 7 in place of weakly holomorphic in the literature. We say φ is weak and write φ Jk,m weakɛa vh b if the support of φ satisfies n 0; φ J k,mɛ a vh b if 4mn r2 0; φ J cusp k,m ɛa vh b if 4mn r2 > 0. The notation J mero k,m χ indicates the vector space of meromorphic functions on H C that transform like a Jacobi form of weight k, index m and multiplier χ. A generalized valuation due to [14] characterizes Jacobi forms from among weakly holomorphic Jacobi forms. Let G = C 0 R/Z p.q. be the additive group of continuous functions g : R R that have period one and are piecewise quadratic. Define the positive non-negative elements in G to be the semigroup of functions whose values are all positive non-negative in R; this makes G a partially ordered abelian group. For φ Jk,m wh χ and x R define ordφ; x = min n + rx + n,r suppφ mx2. The function ord : Jk,m wh χ G, defined by φ ordφ is a generalized valuation in the sense that it satisfies ordφ 1 φ 2 = ordφ 1 + ordφ 2 on the ring of all weakly holomorphic Jacobi forms and ordφ 1 + φ 2 min ordφ 1, ordφ 2 on each graded piece of fixed weight and index. See [14] for the following result: Theorem 3.1. Let φ Jk,m wh χ. We have φ J k,mχ if and only if the valuation is nonnegative, i.e., for all x [0, 1], ordφ; x 0. We have φ J cusp k,m χ if and only if the valuation is positive, i.e., for all x [0, 1], ordφ; x > 0. If m = 0, then ordφ; x is constant. If m > 0, then ordφ; x attains its minimum on 1 2m Z. The following function keeps track of the minimum of the valuation. Definition 3.2. For φ J wh k,m χ, define Ordφ = min 0 x 1 ordφ; x. For a weak Jacobi form φ, all the nonzero values cn, r; φ, with 4mn r 2 0, already occur in the finite initial expansion cn, rq n ζ r. n,r Q: 0 n m/4, r m To prove this, select n 0, r 0 with cn, r = cn 0, r 0 and 4mn r 2 = 4mn 0 r 2 0 and r 0 m as in [7], page 24. The inequality 4mn 0 r implies that 4mn 0 r 2 0 m 2 and hence that n 0 m/4 as well as r 0 m. Thus, to check that a weak Jacobi form is actually a Jacobi form requires the examination of only finitely many Fourier coefficients, namely those up to order m/4 in q. The following dimension formulae are due to Skoruppa and Zagier [36]. Theorem 3.3. Let k, m N. Let σ 0 m be the number of positive divisors of m. Let δk, m be zero unless k = 2 and let δ2, m = 1 2 σ 0m 1 for non-square m and δ2, m = 1 2 σ 0m 1 2 dim J cusp k,m for square m. For even k 2, = δk, m + m j=0 dim S k+2j SL 2 Z j2 4m.

8 8 JEFFERY BREEDING II, CRIS POOR, AND DAVID S. YUEN For odd k 3, dim J cusp k,m m 1 = j=1 dim S k+2j 1 SL 2 Z j2 4m. 4. Theta blocks The theory of theta blocks is due to Gritsenko, Skoruppa, and Zagier, see [14]. Theta blocks are useful for computing bases of spaces of Jacobi cusp forms. Definition 4.1 [14]. A theta block is a function of the form TBf = η fl f0 ϑl η l N for a sequence f : N 0 Z of finite support, where η is the Dedekind eta function ητ = 12 q n2 /24 n n N and we write ϑ l τ, z = ϑτ, lz, where ϑ is Jacobi s odd theta function ϑτ, z = n Z 1 n q 2n+12 8 ζ 2n+1 2. Proposition 4.2 [14]. Let f : N 0 Z have finite support. Then where 2k = f0, L = l N lfl, TBf J mero k,m ɛ K v L H K = f0 + 2 l N fl, 2m = l N l 2 fl. To determine if TBf is a Jacobi cusp form, we have the formula 2 ordtbf, x = k fl 2 B 2 lx, proven in [14], where l N B 2 x = x x 2 x x = n=1 cos2nπx nπ 2. We only consider theta blocks with trivial character, so that K is divisible by 24 and L is even. Moreover, we only consider theta blocks with no theta functions in the denominator, so that we here prefer the notation [l 1,..., l n ] k = η 2k n ϑ l1,, ϑ ln. Due to these restrictions, η 6 ϑ l1 ϑ l2 ϑ l10 will be the form we use for weight two theta blocks, η 3 ϑ l1 ϑ l2 ϑ l9 for weight three, and ϑ l1 ϑ l2 ϑ l8 for weight four. Although theta blocks provide an efficient way to compute Jacobi cusp forms, cuspidal theta blocks do not always span the space. To complete a basis for the space, one might attempt to span other spaces of Jacobi forms and use the index raising and lowering operators. Following a suggestion of D. Zagier, the authors considered another method, which computes bases using only theta blocks of the same index.

9 COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL 9 There are more weak Jacobi forms than cusp forms and so we compute the theta blocks φ whose minimum order Ordφ is not too negative. That is, for a fixed index, we find the theta blocks that have the largest negative minimum order. Sometimes a linear combination of theta blocks from this collection is a Jacobi cusp form. In the cases where we were unable to complete a basis in this manner, we moved onto the collection of theta blocks that had the second largest negative minimum order and tried to complete our basis by taking linear combinations of these theta blocks. This led to more spaces being spanned by linear combinations of theta blocks. If we still didn t have a basis, then we moved on to the collection of theta blocks with the third largest negative minimum order. It turned out that the collection of theta blocks with the third largest negative minimum order was as far as we needed to go to span the spaces of Jacobi cusp forms for weights k = 2, 3, 4 up to the necessary indices. For this information, see the website [3]. Examples of our computations in the weight 2 case are provided below. In each example, we state the dimension of the space, theta blocks in some of the aforementioned collections, graphs of the valuation of the weak theta blocks on the interval [0, 1], linear relations among the listed theta blocks, and a basis in terms of linear combinations of theta blocks. In the cases where we needed to use theta blocks that were weakly holomorphic, we list the terms with nonpositive determinant. In order to search for weight 2 theta blocks, [f 1,..., f 10 ] 2, first note that the index is m = i=1 f i 2. Given the index m, we list all possible ways to write 2m as the sum of 10 squares and then graph the valuation ordtbf; x. This is rigorous because Theorem 3.1 assures us that the minimum is attained on 1 2m Z. We discuss one example in detail because these graphs are informative. Consider j 5 = [1, 1, 2, 3, 4, 4, 4, 5, 8, 14] 2 = ϑ 2 ϑ 2 ϑ 3 ϑ 3 4ϑ 5 ϑ 8 ϑ 14 /η 6. The sum of the squares is 348 and so the index is m = 174 and j 5 J2,174. weak The weak form j 5 is not a Jacobi form and the graph of y = ordj 5 ; x, shown below, dips below the x-axis. The location of these negative minima pinpoint terms in the Fourier expansion of j 5 whose determinant is negative. Recall ordj 5 ; x = minn + rx + mx 2 = 4mn r 2 min n,r suppj 5 4m + mx + r 2m 2. The biggest dip occurs at x = ± 112 2m, corresponding to a value of r = ±112. Evaluating ordj 5 ; = 16 4m tells us that a term is supported with D = 16. From the value of the discriminant 16 = D = 4mn r 2 = 4 174n 112 2, we calculate that n = 18. Indeed, the Fourier expansion of j 5 contains the terms q 18 ζ ±112. The fidelity of the graph is not high enough to see whether or not it dips below the x-axis at x = 59 2m it does, but formula 2 gives ordj 5; ± 59 2m = 1 4m and there is a bad term q 5 ζ 59 with D = = 1. Another inequivalent bad term with D = 1 is given by q 19 ζ 115. In fact, the big dip is actually composed of two separate parabolic arcs meeting at the x-axis.

10 10 JEFFERY BREEDING II, CRIS POOR, AND DAVID S. YUEN y = ordj 5 ; x bad terms: q 5 ζ ±59, q 18 ζ ±112, q 19 ζ ±115. A. Brumer and D. Zagier suggested the Riemann theta relations as a likely source of identities among theta blocks. Consider the identity R 5 from page 20 of [23], ϑτ, z j ϑτ, w j = θ 00 τ, w j θ 01 τ, w j θ 10 τ, w j, j=1 j=1 j=1 where the complex four-tuples z = z 1, z 2, z 3, z 4 and w = w 1, w 2, w 3, w 4 are related by z = wa for the orthogonal matrix A = Proposition 4.3. Let l = l 1, l 2, l 3, l 4 N 4 have l 1 + l 2 + l 3 + l 4 even. Setting p = l 1, l 2, l 3, l 4 A Z 4 and m = l 1, l 2, l 3, l 4 A Z 4, we have ϑτ, l j z + ϑτ, m j z = ϑτ, p j z. j=1 j=1 Proof. The Jacobi theta functions θ 00 τ, z, θ 01 τ, z and θ 10 τ, z are even in z. Therefore setting w equal to l 1 z, l 2 z, l 3 z, l 4 z or to l 1 z, l 2 z, l 3 z, l 4 z gives the right hand side of equation 3 the same value. Therefore we have ϑτ, p j z ϑτ, l j z = 2 ϑτ, m j z + ϑτ, l j z, j=1 j=1 which is equivalent to the conclusion. j=1 j=1 j=1 j=1 j=1

11 COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL 11 Example 4.4. Index 67 dim J cusp 2,67 blocks, namely = 2. There are precisely three cuspidal theta f 1 = [1, 1, 1, 2, 3, 4, 4, 5, 5, 6] 2, f 2 = [1, 1, 2, 2, 3, 3, 4, 4, 5, 7] 2, f 3 = [1, 1, 1, 2, 2, 3, 3, 4, 5, 8] 2. f 1 f 2 f 3 The graphs y = ordf i ; x of their valuations on [0, 1] show, by staying strictly above the x-axis, that the f i are cusp forms. Each of these blocks is in the span of the other two by the linear relation f 1 f 2 + f 3 = 0. The dimension formula in Theorem 3.3 makes the verification of such identities trivial and so a basis for J cusp 2,67 is {f 1, f 2 }. Alternatively, this identity is a consequence of Riemann s theta relation. Six of the ten entries for each of the f i are common and the remaining four-tuples satisfy the relation [2, 3, 4, 7] 2 = [1, 2, 3, 8] 2 + [1, 4, 5, 6] 2, which follows from Proposition 4.3 by taking l = 8, 3, 2, 1, for instance. Example 4.5. Index 191 dim J cusp 2,191 = 2. There is precisely one cuspidal theta block, namely f 1 = [1, 2, 3, 3, 4, 5, 6, 7, 8, 13] 2. The graph y = ordf 1 ; x of its valuation on [0, 1] is the following.

12 12 JEFFERY BREEDING II, CRIS POOR, AND DAVID S. YUEN In this case, the cuspidal theta blocks do not produce a basis. So we consider the theta blocks with the largest negative minimum order, which is 5/764. The theta blocks with this order, graphs of their valuation on [0, 1], and the offending terms that prevent them from being cuspidal are below. g 1 = [1, 1, 3, 4, 5, 6, 7, 8, 9, 10] 2, g 2 = [1, 3, 3, 4, 4, 5, 7, 7, 8, 12] 2, g 3 = [1, 2, 3, 3, 3, 5, 6, 8, 9, 12] 2, g 4 = [1, 1, 2, 3, 3, 4, 5, 5, 6, 16] 2. We have the following linear relations f 1 g 1 + g 2 = 0, f 1 + g 3 + g 4 = 0. A basis for J cusp 2,191 is {f 1, g 1 g 4 }. The linear relations again follow from Riemann s theta relations. The first reduces to [3, 4, 7, 12] 2 = [2, 3, 6, 13] 2 +[1, 6, 9, 10] 2 and the second to [4, 7, 8, 13] 2 = [1, 4, 5, 16] 2 +[3, 8, 9, 12] 2. These follow from Proposition 4.3 regardless which of the three occurring four-tuples is selected to be l. The result that {f 1, g 1 g 4 } is a basis requires that g 1 g 4 be proven to be a cusp form. Since g 1 g 4 is not a theta block but a linear combination of theta blocks of index m = 191, we have little choice but to compute the Fourier expansion of g 1 g 4 to order q m/4 and check the positivity of the determinant for each term. g 1 g 2 bad terms: q 41 ζ ±177. bad terms: q 41 ζ ±177. g 3 g 4 bad terms: q 41 ζ ±177. bad terms: q 41 ζ ±177. Example 4.6. Index 174 dim J cusp 2,174 = 1. There are no cuspidal theta blocks. The largest negative minimum order is 1/174. There is precisely one theta block with this order, namely g 1 = [1, 2, 3, 4, 4, 5, 7, 8, 8, 10] 2. Since we only have one such theta block, we move on to the collection of blocks with the second largest negative minimum order, which is 3/232. The theta blocks

13 COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL 13 with this order are: h 1 = [1, 2, 3, 4, 5, 5, 7, 7, 7, 11] 2 ; h 2 = [1, 1, 2, 3, 3, 3, 4, 5, 7, 15] 2. The linear combination g 1 h 1 + h 2 that cancels the bad coefficients is zero, so we move on to the collection of blocks with the third largest negative minimum order, which is 2/87. The theta blocks with this order, graphs of their valuations on [0, 1], and the offending terms that prevent them from being cuspidal are below. j 1 = [2, 2, 2, 3, 5, 5, 7, 8, 8, 10] 2 ; j 2 = [1, 1, 2, 2, 5, 6, 7, 8, 8, 10] 2 ; j 3 = [1, 2, 4, 4, 4, 5, 6, 7, 8, 11] 2 ; j 4 = [1, 1, 1, 2, 3, 4, 5, 7, 11, 11] 2 ; j 5 = [1, 1, 2, 3, 4, 4, 4, 5, 8, 14] 2. The graph of ordj 5 ; x has already been discussed in detail. We have the following linear relations amongst the theta blocks in these collections and a basis for this space is {j 1 + j 5 }. g 1 h 1 + h 2 = 0; [5, 7, 7, 11] 2 = [1, 3, 3, 15] 2 + [4, 8, 8, 10] 2, g 1 j 1 + j 2 = 0; [2, 2, 3, 5] 2 = [1, 1, 2, 6] 2 + [1, 3, 4, 4] 2, h 1 j 3 + j 4 = 0; [4, 4, 6, 8] 2 = [1, 1, 3, 11] 2 + [3, 5, 7, 7] 2, g 1 j 3 + j 5 = 0; [4, 6, 7, 11] 2 = [1, 3, 4, 14] 2 + [3, 7, 8, 10] 2. j 1 j 2 bad terms: q 5 ζ ±59, q 18 ζ ±112, bad terms: q 18 ζ ±112, q 20 ζ ±118 q 19 ζ ±115 j 3 j 4 bad terms: q 18 ζ ±112, q 20 ζ ±118 bad terms: q 18 ζ ±112, q 19 ζ ±115, q 42 ζ ±171, q 43 ζ ±173

14 14 JEFFERY BREEDING II, CRIS POOR, AND DAVID S. YUEN 5. Formal Fourier-Jacobi expansions The Fourier expansion of a paramodular form f M k KN is of the form fz = T at ; fe Z, T. Here we set A, B = trab for symmetric matrices A, B. The summation is over semidefinite T with 2T even and T, divisible by N. When f is a cusp form, then the summation is over T that are also definite and we use the notation X 2 N = { n r/2 > 0 : n, r, m Z} for the summation indices. r/2 Nm The group ˆΓ 0 N is the transpose of the group ˆΓ 0 N. For U ˆΓ 0 N, we have detu k au T U; f = at ; f because uˆγ 0 N KN and the action of ˆΓ 0 N naturally stabilizes X 2 N via T T [U] = U T U. By writing elements of H 2 as Z = τ z ω z and by expanding the Fourier series in terms of ξ = eω, one has fz = φ Nm τ, zξ Nm, m=0 cn, r; φ Nm = a n r/2 r/2 Nm where each φ Nm is a Jacobi form of weight k and index Nm. If f is a cusp form, then the φ Nm are all Jacobi cusp forms. We now explain how to implement our method for computing S k KN, χ. Suppose we know that the space S k KN, χ is determined by its first l Fourier- Jacobi coefficients. Section 7 will show how to calculate such l without already knowing the dimension of S k KN, χ. Let V = l j=1 J cusp k,nj. The map FJ l : S k KN, χ V that sends a paramodular form to its first l Fourier-Jacobi coefficients is injective and each paramodular Atkin-Lehner involution µ α provides equations satisfied by the image of S k KN, χ under FJ l. Proposition 5.1. For α N, fix x, y Z such that αy N α x = 1. Let ɛ α = ±1. Assume that f S k KN satisfies f µ α = ɛ α f and let j=1 φ jnξ jn be the Fourier-Jacobi expansion of f. For all n, r, m Z we have 4 ɛ α cn, r; φ Nm = c αn + xr + x 2 N α m, 2Nn + αy + N α xr + 2xyNm; φ N N α n+yr+y2 αm. Proof. Define U by setting U = 1 N αy, the paramodular Atkin-Lehner involution for α is µ α = uu. We calculate ɛ α cn, r; φ Nm = ɛ α a ; f = at ; f µ α, where we set T = n r/2 r/2 Nm α α x n r/2 r/2 Nm. Furhermore, using detu = 1, ; f at ; f µ α = at ; f uu = detu k au T U 1 ; f = au T U 1 ; f. Multiplying the three matrices gives U T U 1 αn + xr + x = 2 N α m Nn + αy r 2 + N α x r 2 + xynm, Nn + N α x r 2 + αy r 2 + xynm N α n + yr +. y2 αmn Therefore we have au T U 1 ; f = c αn + xr + x 2 N α m, 2Nn + αy + N α xr + 2xyNm; φ N N α n+yr+y2 αm.

15 COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL 15 For a fixed Atkin-Lehner character χ : KN {±1}, denote by V χ the elements φ N, φ 2N,..., φ ln of V that satisfy equation 4 for all α N with ɛ α = χµ α and αn+xr+x 2 N α m l and N α n+yr+y2 αm l. The image of S k KN, χ under the injection FJ l is contained in V χ and dim S k KN, χ dim V χ. In section 4 many spaces of Jacobi cusp forms were spanned by theta blocks. Theta blocks have integral Fourier coefficients and the equations defining V χ are of a simple type: a Fourier coefficient of one Jacobi form equals plus or minus a Fourier coefficient of another Jacobi form. We may compute dim V χ as the corank of a large integral matrix Mχ with dim V columns and a row for every applicable equation from 4. It can be important for run time to note that rank Q Mχ rank Fp Mχ so that we may compute the rank of Mχ modulo a prime p and still have dim S k KN, χ corank Fp Mχ. Finally, if one can construct χ corank F p Mχ linearly independent cusp forms f i S k KN, then one has proven that S k KN = Span C f i and that FJ l : S k KN, χ V χ is an isomorphism for each χ. It is the proof of a correct upper bound that is usually the more difficult aspect; modular forms can be constructed in a great variety of ways and this has always been part of the charm of the subject. In this context, the most obvious techniques for constructing cusp forms are Gritsenko lifts, traces of theta series and Borcherds products. Constructions in low weights are typically the more challenging. We give one nontrivial example. The first nonlift for weight four occurs at level N = 31 but this is covered by Ibukiyama s dimension formula because N is prime. The first nonlift that occurs at a non-squarefree level is at N = 40 and we illustrate its construction. Our upper bound for dim S 4 K40 is 5 and the space of lifts is only four dimensional. Let Ξ J cusp 4,80 be given by Ξ = [1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5] 4. Let V 2 : J k,m J k,2m be the index doubling Hecke operator from [7], page 41. Let φ J cusp 4,40 be given by φ = [1, 1, 1, 2, 4, 4, 4, 5] 4. Let ψ J wh 0,40 be defined by ψ = Ξ φ V 2 /φ. The weakly holomorphic Jacobi form ψ has integral Fourier coefficients and, by the theorem on page 29 of [9], gives a holomorphic Borcherds product Borchψ S 4 K40. The Fourier-Jacobi expansion shows that Borchψ is not a Gritsenko lift and thus that dim S 4 K40 = 5. See [32] for such constructions. 6. Vanishing theorems We prove vanishing theorems for paramodular cusp forms. For prime levels these results were first given in [30]. We review the results of [26] [28] [29]. Let P n R and Pn semi R be the spaces of positive definite and semidefinite n-by-n symmetric real matrices, respectively. Definition 6.1. A function φ : Pn semi R R 0 is called type one if For all s P n R, φs > 0, for all λ R 0 and s Pn semi R, φλs = λφs, for all s 1, s 2 Pn semi R, φs 1 + s 2 φs 1 + φs 2. Type one functions are continuous on P n R and respect the partial order on R. Basic examples are: For s P semi R, define P semi n ms = inf u Zn\{0} u su, the Minimum function, trs = inf u GLnZ tru su, the reduced trace, δs = dets 1/n, the reduced determinant, n

16 16 JEFFERY BREEDING II, CRIS POOR, AND DAVID S. YUEN ws = inf u,s u PnR mu, the dyadic trace. For n = 2, the dyadic trace of a Legendre reduced s = a b b c P2 R is given by ws = a + c b, see [25]. Legendre reduced means 2 b a c. For f M k Γ, we set suppf = {T P n Q : at ; f 0}. Theorem 6.2. Vanishing Theorem for general subgroups. Let φ be type one. For all n N there exists a c n φ R >0 such that: For any subgroup Γ Sp n Z with finite index I and coset decomposition Sp n Z = I i=1 ΓM i, we have k N, f S k Γ, 1 I I inf φ suppf M i > c n φ k = f 0. i=1 For n = 2, we may take c n φ = inf φ [ ]. Proof. This is Theorem 2.5 from [29], except for the last comment, which is Corollary 5.8 from [26]. We can apply this theorem to Γ = Γ 0N Sp 2 Z and use the double coset decomposition of section 8 to get a vanishing theorem for cusp forms in S k Γ 0N, which also applies to paramodular forms because Γ 0N KN. We begin with some lemmas. For any α N with α N, denote w α = N α 0 1 Even though the single cosets representatives of Γ 0N\Sp 2 Z are enumerated with some determination in section 8, we only require the following lemma here. Recall, φn = Z/nZ, ψn = {x Z/nZ : x 2 = 1} and ϖn = φn/nψn. Lemma 6.3. Let α, N N with α N and set γ = α, N α. We have the disjoint single coset decomposition for some u i P 2,0 Z and for Γ 0Nw α P 2,0 Z = κ α = α 2 ϖγ N κ α i=1. prime p N Γ 0Nw α u i p. As α ranges over α N, we have distinct double cosets Γ 0Nw α P 2,0 Z. Proof. The first assertion is Lemma 8.19 and the second is Lemma Remark 6.4. The significance of using double cosets where the right hand group is P 2,0 Z is that when u i P 2,0 Z, and when φ is a class function, then κ φsuppf wu i = κ φsuppf w. i=1 Lemma 6.5. For α, N N with α N, we have w α = = KN N α α 0 1 N N α 0 1 N α α 0 1 N α.

17 COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL 17 Lemma 6.6. Let f S k KN be a cusp form. Then suppf w α = 1 α 0 α α α suppf 1 0 α. Proof. Since f S k KN, then by Lemma 6.5, and the latter has support 1 0 f w α = f 0 1 α suppf α 0 1 N α 1 0, 0 1 α α and the result follows. α Note that for T suppf, the matrix α T remains halfintegral and of the same determinant. Here is our first vanishing theorem. α N α Theorem 6.7 Vanishing Theorem I for paramodular cusp forms of arbitrary level. Let f S k KN. Let φ be a type one GL 2 Z-class function. If f 0, then α ϖα, N α inf φ α 0 α α suppf 1 0 α φ kn 2 prime q N q 2. Proof. Set Γ 2 = I i=1 Γ 0NM i with I = Γ 0N\Γ 2 = N p p but 2 use only the subset of representatives given by Γ 2 α N Γ 0Nw α P 2,0 Z = κα α N i=1 Γ 0Nw α u α,i, for u α,i P 2,0 Z. These single cosets are distinct by Lemma 6.3. We plug these distinct single cosets into Theorem 6.2 to get α 2 ϖα, N α N q inf φ suppf w α α N prime q N φ kn 3 prime q N The result follows from Lemma 6.6 and some simplification q q 2. We remark that the single cosets used in the proof of Theorem 6.7 are all from the paramodular identity zero-dimensional cusp KNP 2,0 Q. Corollary 6.8. Let f S k KN. Let δ be the reduced determinant function. If f 0, then 2 inf δ suppf k 1 α ϖα, N 15 α N q. 2 α N prime q N Proof. This follows from Theorem 6.7 using the reduced determinant function. Some cosets can be represented by the paramodular Atkin-Lehner involutions µ α 1 defined in section 2. Denote ρ α = u α I 2 and note at ; f ρ α = α k aαt ; f so that suppf ρ α = 1 α suppf. Lemma 6.9. Let α, N N with α N such that α, N/α = 1. Then w α KNµ α ρ α P 2,0 Z.

18 18 JEFFERY BREEDING II, CRIS POOR, AND DAVID S. YUEN Proof. Let h Z be such that hα 1 mod N α. We can directly verify that w α = α 0 hα 1 N N 0 h µ α ρ α x 2α t 1 N α 0 t 1hα 1 α1 hαx t 2 x N 0 0 x 1 t 2 t N α x 2α Theorem 6.10 Vanishing Theorem II for paramodular cusp forms of arbitrary level. Let f S k KN. Let φ be a type one GL 2 Z-class function. If f 0, then α inf φ suppf µ α α N:α, N α =1 + α ϖα, N α inf φ α N:α, N α 1 α α suppf. α α φ kn 2 prime q N 1 + q 2. Proof. For each α N with α, N/α = 1, we have by Lemma 6.9 that f w α = f µ α ρ α u for some u P 2,0 Z. Since φ is a class function, then φsuppf µ α ρ α u = φsuppf µ α ρ α Thus φ 1 α α α suppf Plugging this into Theorem 6.7 proves the result. α α = φsuppf w α = 1 α φsuppf µ α. In practice, we might either ignore the terms where α, N α 1 or replace those terms by some constant lower bound. A simple corollary is the following. Corollary Let φ be a type one GL 2 Z-class function. If f 0, then α inf φ suppf µ α φ kn q 2. α N:α, N α =1 prime q N Theorem 6.12 Vanishing Theorem III for paramodular cusp eigenforms of arbitrary level. Let φ be a type one GL 2 Z-class function. Let f S k KN be an eigenform for the involution µ α for every α N with α, N α = 1. Then f 0 implies inf φsuppf φ k N q r N q r + q r 2 q r + 1. Proof. This follows from Corollary 6.11 and by noting that α = q r + 1. α N:α, N α =1 q r N 7. Determining numbers of Fourier-Jacobi coefficients The determining sets of Fourier coefficients that were worked out in the previous section can be used to give upper bounds for the number of Fourier-Jacobi coefficients that determine a paramodular cusp form. The most direct approach is to simply count the number of Fourier-Jacobi coefficients needed to cover the Fourier coefficient indices required by the determinant bound of Corollary 6.8 or the bound for general type one class functions and Atkin-Lehner eigenforms of Theorem To this end we make the following definitions.

19 COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL 19 Definition 7.1. Let N N. For T X 2 N, define m N T, m + N T and m N T to be the minimum of 1 N gt g, over g ˆΓ 0 N, Γ + 0 N and Γ 0N, respectively. For λ R + and a type one function φ, define J N φ, λ, J + N φ, λ and J N φ, λ to be the maximum of m N T, m + N T and m N T, respectively, over T X 2N satisfying φt λ. We note that these minima exist and are constant on ˆΓ 0 N-orbits of X 2 N. We do not address the existence of the maxima except in specific cases and in principle allow + as a maximum. Theorem 7.2. Let f S k KN have a Fourier-Jacobi expansion given by f = j=1 φ Njξ Nj. If φ Nj = 0 for j J N δ, δ o then f = 0, where δ o = k 2 15 N 2 prime q N q 2 α N α ϖα, N α. For f S k KN ±, we have f = 0 if φ Nj = 0 for j J + N δ, δ o. For Atkin-Lehner eigenforms f S k KN, χ, we have f = 0 if φ Nj = 0 for j J N δ, δ o. Proof. By Theorem 6.8 it suffices to prove that at ; f = 0 for T X 2 N with δt δ o. Since δt δ o, we have m N T J N δ, δ o and g T g = for m = m N T and some g ˆΓ 0 N and n, r Z. Thus detg k at ; f = a ; f = cn, r; φ Nm = 0 n r/2 r/2 Nm by m J N δ, δ o and the hypothesis. The other two cases are similar. n r/2 r/2 Nm Theorem 7.3. Let φ be a type one function that is a GL 2 Z-class function. Let f S k KN, χ have Fourier-Jacobi expansion f = j=1 φ Njξ Nj. If φ Nj = 0 for j JN φ, λ then f = 0, where λ = φ k N q r + q r 2 q r + 1. q r N Proof. The proof is the same as the proof of Theorem 7.2 except we use Theorem 6.12 in place of Corollary 6.8. The above two theorems are satisfactory when circumstances permit running a computer program to tabulate J N δ, δ o or JN φ, λ. We next formulate upper bounds of theoretical interest in terms of the Jacobsthal function defined in the Introduction. Indeed, the upper bounds given here are further motivation for studying the growth of the Jacobsthal function jn. H. Iwaniec proved that jn Oln N 2, see [22]. The Jacobsthal function is labeled A in the Online Encyclopedia of Integer Sequences. We begin with the following lemma. Lemma 7.4. Let N = p α1 1 pα l l, where p i p j for i j and α i 1, be the prime factorization of N. Set Ñ = p 1 p l and let µ N satisfy the condition: 2µ + 1 jñ/p i for all i. Then, for all A, B, N N with A, B = 1, we have B, N = 1 or there exists a y Z with y µ such that A + By, N = 1. Proof. Suppose that B, N 1 and, by rearranging the prime factors of N, suppose that p l r, p l r+1,..., p l B but p 1,..., p l r 1 B, where r 0. Consider j satisfying 1 j l r 1. If p j A + y 1 B and p j A + y 2 B, then p j y 1 y 2 B,

20 20 JEFFERY BREEDING II, CRIS POOR, AND DAVID S. YUEN which implies that p j y 1 y 2 since p j B for 1 j l r 1. Suppose by way of contradiction that there does not exist a y with y µ such that p 1 p l r 1, A + yb = 1. That is, suppose for all y µ that there exists p j A + yb for 1 j l r 1. Since B, p 1 p l r 1 = 1, there exists z such that A + zb is a multiple of p 1 p l r 1. Consider the 2µ + 1 consecutive numbers µ z, µ 1 z,..., z,..., µ 1 z, µ z. Ñ Ñ Since 2µ + 1 j j = jp 1 p l r 1, then at least one of p l p l r p l these is relatively prime to p 1 p l r 1, call that y z, where y µ. For each 1 j l r 1, since p j A+zB and p j y zb, then p j A+zB +y zb, which is p j A + yb. Thus p 1 p l r 1, A + yb = 1. Thus in any case, there exists a y with y µ such that p 1 p l r 1, A + yb = 1. Fix this y. Now, for l r j l, then p j B implies p j A, which implies p j A + yb. Thus p j A + yb for all 1 j l. That is, A + yb, N = 1. In order to transfer our estimates from Fourier coefficients to Fourier-Jacobi coefficients we bound m + N from above by a type one GL 2Z-class function. Lemma 7.5. Let N N. Choose µ N to satisfy the condition in Lemma 7.4. For T X 2 N, we have m + N T mt, m + N T < trt, if N is prime, if N is a prime power, m + N T µ + µ2 trt, in general. A B Proof. Let T X 2 N. There exists a σ = GL 2 Z such that T [σ] is a b Legendre reduced. Thus we may assume that T [σ] = with a c 2b 0, b c if we choose the reduction conditions in this manner. In this version, c = mt. Consider the case where N = p is prime. If p B then σ ˆΓ 0 N and m N T = a b m N = c/n c = mt. If B, p = 1 then choose β Z satisfying βb 1 b c A B 0 1 mod p and note that 1 Aβ ˆΓ 0 N. Therefore T is in the same ˆΓ 0 N-orbit as [ ] [ ] [ ] A B 0 1 a b 0 1 c T = =. 1 Aβ b c 1 Aβ c Noting FN F 1 N =, we have m + Nc N T c = mt. Now consider the case where N = p r is a prime power, so that B, N = 1 or A, N = 1. If B, N = 1, we choose β as before and get m + N T c < c + a = A B 1 y trt. If A, N = 1, then for some choice of y Z we have 0 1

21 COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL 21 ˆΓ 0 N and T is in the same ˆΓ 0 N-orbit as [ ] [ ] [ ] A B 1 y a b 1 y a T = =. 0 1 b c 0 1 Therefore, m + N T a < trt. In the case of general N we use our hypothesis on µ and Lemma 7.4 to obtain B, N = 1 or the existence of a y Z with y µ and A+By, N = 1. If B, N = 1 we proceed as before and obtain m + N T c < trt < µ+µ2 trt, making use of µ > 0. For the main case, suppose A+By, N = 1 so that αa+by = 1+lN 1 αb for some α, l Z. Set S = SL y αa ln 2 Z and check that A B 1 αb A + By αbln σs = = y αa ln α ˆΓ 0 N. Then T is in the same ˆΓ 0 N-orbit as [ ] a b 1 αb a 2by + cy 2 T [σs] = =, b c y αa ln and therefore m + N T a 2by + cy2. If y = 0, we have m + N T a 2by + cy2 = a < a + c < µ + µ2 trt. If y 0, we argue a 2by + cy 2 = a + c 2 a + c 2 2by + a + c 2 y2 + a c 2 1 y2 + a + c 2 y + a + c 2 y2 + 0 = 1 2 a + c1 + y + y trt 1 + µ + µ 2. Theorem 7.6 proves Theorem 1.2 of the Introduction. Theorem 7.6. Let f S k KN, χ have Fourier-Jacobi expansion f = φ jn ξ jn. j=1 Let N = p α1 1 pα l l be the prime factorization of N with α i N and distinct primes p i and set Ñ = p 1 p l. Choose µ N such that 2µ + 1 jñ/p i for all i. Let κ be 1 when N is prime, 2 when N is a composite prime power and 1 + µ + µ 2 otherwise. If φ jn = 0 for j κ k 10 N p r + p r 2 p r, then f = 0. When N is a + 1 p r N composite prime power this inequality may be taken strictly. Proof. As in Theorem 7.2, this follows from the Fourier coefficient bound for the reduced trace tr and from the bound m + N T 1 2 κ trt from Lemma 7.5

22 22 JEFFERY BREEDING II, CRIS POOR, AND DAVID S. YUEN Table 5. Examples of bounds on the number of Fourier-Jacobi coefficients. k N Jstl J N δ, δ o J + N δ, δ o JN δ, δ o JN tr, λ JN w, λ MIN Theorem 7.6 should be viewed as a worst case scenario, the most theoretical among a similar group of theorems all proven by reduction to a determining set of Fourier coefficients. If we use Iwaniec s bound on the Jacobsthal function, the required number of Fourier-Jacobi coefficients in Theorem 7.6 is O knlnn 4. Various bounds on a determining number of Fourier-Jacobi coefficients are given in Table 5. The column headed J N δ, δ o gives a number of Fourier-Jacobi coefficients sufficient to determine S 2 KN; the column J + N δ, δ o determines S 2 KN ± ; and the columns: Jstl, J N δ, δ o, J N tr, λ and J N w, λ determine S 2KN, χ. In this last case, there are examples of each of the three type one functions δ, tr and w winning over the other two. A more extensive list can be seen at [3]. 8. Coset representatives of Γ 0N\Sp 2 Z/P 2,0 Z This section provides detailed coset representatives for Γ 0N\Sp 2 Z/P 2,0 Z and for Γ 0N\Sp 2 Z. We briefly mention Satake compactifications and cusps as they pertain to this article, compare [2][35]. For Γ Sp 2 Q commensurable with Sp 2 Z, let SΓ\H 2 be the Satake compactification of Γ\H 2. The one-dimensional cusps of SΓ\H 2 correspond bijectively with the double cosets of Γ\Sp 2 Q/P 2,1 Q and the zero-dimensional cusps of SΓ\H 2 correspond bijectively with the double cosets of Γ\Sp 2 Q/P 2,0 Q. In [34], Reefschläger classified the double cosets of KN\Sp 2 Q/P 2,1 Q and the following is an easy corollary. Theorem 8.1. Reefschläger Let N N. We have a disjoint union Sp 2 Q = KN P 2,1 Q, m N: m N 1 m m 1 In particular, the number of 1-dimensional cusps is τn. The zero-dimensional cusps of KN were enumerated in [31] by finding all the zero-dimensional cusps of each one-dimensional cusp of KN. Theorem 8.2 Theorem 1.3, [31]. Let N, f, N 0 N with N = f 2 N 0 for N 0 squarefree. We have a disjoint double coset decomposition Sp 2 Q = KNC 0 McP 2,0 Q, with C 0 x = x 1 0, c, M x c, M Z/cZ /{±1}, c N: c f

23 COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL 23 where M is prime to N and gives the class of M in Z/cZ /{±1}. The number of 0-cusps is 1 + f/2. It is known, see [12], that the index of Γ 0N in Sp 2 Z is Γ 0N\Sp 2 Z = N q + 1 q q 3. prime q N We indicate the proof. Recall that P 3 Z/NZ consists of equivalence classes of relatively prime 4-tuples of integers, where [α i ] [β i ] means that there is an integer m prime to N with mα i β i mod N for all i = 1, 2, 3, 4. Sending the bottom row of an integral symplectic matrix to its projective class in P 3 Z/NZ identifies the cosets in Γ 0N\Sp 2 Z with this finite projective space. To see that this map is injective, we note the following lemma, which will be used again. Lemma 8.3. Let σ Sp 2 Z. Then σ Γ 0N if and only if σ 12, σ 32, σ 42 are all multiples of N or σ 41, σ 42, σ 43 are all multiples of N. Proof. Assume that σ = C A D B Sp 2Z satisfies A 12, C 12, C 22 0 mod N. We need to show that C 21, D 21 0 mod N. The symplectic condition that A C is symmetric gives us A 22 C 21 0 and A D C B = I 2 gives us A 22 D It now suffices to show that A 22 is prime to N and this follows from our hypothesis and detσ = 1. Using CD symmetric and AD BC = I 2 will prove the alternate version. This bottom row map is surjective because any primitive element of Z 4 may be a selected row of an integral symplectic matrix. To count the number of elements in P 3 Z/NZ, first consider the case where N = q r. The number of vectors in Z/q r Z 4 not all divisible by q is N q. Dividing this by Z/q r Z = q r q yields the answer N q + 1 q q. The index for general N follows by the 3 Chinese Remainder Theorem. Simpler but also useful is the disjoint union 5 KN = Γ N 0 a b Γ c d 0 N\SL 2Z Γ 0Ni 2 a b/n Nc d This shows Γ 0N\KN = Γ 0 N\SL 2 Z = P 1 Z/NZ = N p N p. We also indicate the proof. For σ KN, we know that g = gcdn, σ 22 and α = Nσ 24 are relatively prime because N N/g N σ 1 = detσ = det 22 α/n σ = det 22 /g α. N N/g N N N N g g g In particular, for σ Γ 0N, σ 22 is relatively prime to N. This allows us to define a map Γ 0N\KN P 1 Z/NZ by sending σ to [σ 22 +ln, Nσ 24 ] for any l Z making σ 22 + ln and Nσ 24 relatively prime. Equation 5 now follows because the representatives are inequivalent. Our strategy to create detailed coset representatives for the zero-dimensional cusps of Γ 0N is to first find representatives associated to each zero-dimensional.