Maass relations for Saito-Kurokawa lifts of higher levels

We use an automorphic approach to Saito-Kurokawa lifts to derive Maass relations for lifts of classical cuspidal modular forms of arbitrary level.


Introduction
It is known that a Siegel modular form F is a (classical) Saito-Kurokawa lift of an elliptic modular form f if and only if its Fourier coefficients satisfy the Maass relations ) .
The classical cuspidal Saito-Kurokawa lift of weight k is a lift from a cuspidal modular form f ∈ S (1) 2k−2 (SL 2 (Z)) with k even; it is a cuspidal Siegel modular form F ∈ S (2) k (Sp 4 (Z)). The first construction of such a lift was given by Maass in [11] using correspondences between Siegel and classical modular forms, Jacobi forms and modular forms of half-integral weight (see also [5]). However, Saito-Kurokawa lifts can be also constructed using representation theory ( [14], [24]). The advantage of the latter is that it can be easily generalised to lifts of modular forms of higher level, and also with an odd weight. In this case, if k is even 1 , for any f ∈ S (1) 2k−2 (Γ 0 (N )) we get a cuspidal Siegel modular form of weight k invariant under the action of a congruence subgroup of GSp 4 (Z) such that its spin L-function is given by L(s, F ) = L(s, f )ζ(s − k + 1)ζ(s − k + 2). This does not tell us though anything about the coefficients of F and whether they satisfy similar Maass relations. Pitale, Saha and Schmidt showed in [15] that this is indeed the case if F ∈ S (2) k (Sp 4 (Z)) is a Hecke eigenform. From a representation theoretic point of view, a Saito-Kurokawa lift produces from a cuspidal automorphic representation π of PGL 2 (A) a cuspidal automorphic representation Π of PGSp 4 (A); we can think of f and F as vectors of matching weight in the vector spaces of π and Π. What is important is that any representation Π we obtain via this (generalised) Saito-Kurokawa lifting is a CAP representation.
More precisely, consider a cuspidal Siegel modular form F of level Γ 0 (N 1 , N 2 ). We say that F is associated to a CAP representation if the following are true.
1) The adelisation of F gives rise to an irreducible automorophic representation Π of GSp 4 (A). 2) The representation Π is equivalent at almost all places to a constituent of a globally induced representation from a proper parabolic subgroup of GSp 4 .
Furthermore, we say that F is associated to a P-CAP representation if the proper parabolic subgroup above is the Siegel parabolic subgroup. The classical Saito-Kurokawa lifts correspond exactly to the P-CAP representations. It is known that if k ≥ 3, then F that is associated to a CAP representation is automatically associated to a P-CAP representation. If k = 1 or 2, one also has CAP representations associated to other parabolics (the so-called B-CAP and Q-CAP representations). Note that the first condition above automatically implies that F is an eigenform of the local Hecke algebra at all primes not dividing N 2 . For general N 1 , N 2 , there is no known explicit construction that generalises the classical Saito-Kurokawa lifts and exhausts the set of all P-CAP F of level Γ 0 (N 1 , N 2 ). It seems difficult then to directly prove the Maass relations from construction. In this work we are able to prove the Maass relations using methods of representation theory.
Theorem. Let N 1 , N 2 be positive integers, N 1 |N 2 , and let F be a cuspidal Siegel modular form of weight k and level Γ 0 (N 1 , N 2 ) that is associated to a P-CAP representation. Let a, b, c be integers such that gcd(a, b, c, N 2 ) = 1, b 2 − 4ac < 0 and ).
In fact, theorem stated above is a corollary of a more general result (Theorem 6.1), where the condition on b 2 −4ac p is replaced with an exact condition on a certain ray class group. The simplification presented above follows from a connection between equivalence classes of binary quadratic forms and elements of suitable ray class groups; this is the topic of section 4 and may be of independent interest.
A reader may find it surprising or unsatisfactory that, in contrary to the classical Maass relations, the right hand side of the above equality contains L. However, there are modular forms for which this cannot be improved. Indeed, as Schmidt showed in [23], there exist paramodular forms that are Saito-Kurokawa lifts, and all their Fourier coefficients a(F, T ) have of course the (2, 2)-entry of T divisible by the level.
We should also mention that a relation similar to the one above (without L on the right) was considered by Ibukiyama in [7]. However, his Saito-Kurokawa lift was defined on the space of Jacobi forms of index 1 and it was assumed that such a Jacobi form occurs in a Fourier Jacobi expansion of Siegel modular forms in the image of the lift.
Our work extends the method developed in [15] to Saito-Kurokawa lifts of higher levels. It bases on the relation satisfied by vectors in local Bessel models for P-CAP representations ( [15], Theorem 2.1) and the fact that certain values of a global Bessel period associated to a Siegel modular form F can be expressed in terms of Fourier coefficients of F . We compute these values explicitly for Siegel modular forms invariant under the action of Γ 0 (N 1 , N 2 ) with N 1 |N 2 , and combine it with the local-global relation (11) satisfied by Bessel periods to obtain a relation between Fourier coefficients (Theorem 5.1; generalisation of [10, Theorem 2.10]). Theorem 5.1 is a basis for our main result.
Throughout the paper we use the following notation.
• N, Z, Q, R, C stand for the natural, integer, rational, real and complex numbers respectively; Q p denotes the p-adic numbers and Z p the p-adic integers, A stands for the adeles of Q and A f := ′ p<∞ Q p the finite adeles; the set of invertible elements in a ring R is denoted by R × ; • M n denotes the set of n × n matrices, whose identity element is 1 n , and i 2 := i1 2 ; we use the superscript M sym n for symmetric matrices, and M + n for the matrices with positive determinant; we distinguish a set where half-integral means that T has integers on the diagonal; t T is the transpose of T and tr T the trace of T ; are the content and the discriminant of the matrix The letter G will always stand for the group GSp 4 defined as follows: ; and for n 1 ≤ n 2 : • For N = p p np and N 1 |N 2 having similar prime factorizations we define I N1,N2 := p<∞ I(n 1,p , n 2,p ); where ord p X := max{n ∈ Z : p n |X}, and N ∞ denotes a formal number such that N l |N ∞ for all l ∈ Z + ; denotes the Legendre symbol.

Acknowledgements
The work presented in this paper was carried out at the University of Bristol and represents a part of PhD thesis of the author. Her studies and research were possible thanks to a funding provided by EPSRC. The author would like to thank her supervisor Abhishek Saha for guidance, support and patience in explaining various subtleties.

Preliminaries
Let π = ⊗ p π p be an irreducible automorphic cuspidal representation of G(A) with trivial central character and such that π ∞ = L(k, k), the lowest weight representation of scalar minimal K-type of weight k. Let Φ be an automorphic form in the space of π and let φ ∞ be a lowest weight vector of π ∞ . This means that Define where g ∈ G(R) is such that g · i 2 = Z and

Such a function F is holomorphic and satisfies
; it is a cuspidal Siegel modular form of degree 2, level Γ 0 (N 1 , N 2 ) and weight k that is an eigenform of the local Hecke algebra at all primes p ∤ N 2 . 2 It follows that F admits a unique Fourier expansion a(F, T )e(tr (T Z)), where e(x) := e 2πix , and its Fourier coefficients satisfy Observe that the correspondence (3) is bijective in a sense that to any Siegel cusp form F that satisfies the above conditions and gives rise to an irreducible representation we can attach an automorphic form Φ via Φ is called the adelisation of F . 2 In fact, these functions, coming from irreducible representations, span the space of Siegel cusp forms of degree 2, level Γ 0 (N 1 , N 2 ) and weight k.

Local and global Bessel models
Throughout this section F is a non-archimedean local field of characteristic zero, ø its ring of integers, p the maximal ideal of ø, ̟ a generator of p, and q the cardinality of the residue field ø/̟ø. For our global application we will only need F = Q p , but because of a deeper analysis of an ideal class group in the next section, we would like to describe a torus T in more detail.
3.1. Local Bessel models for GSp 4 . We recall the definition of the Bessel model following the exposition of Furusawa [6] and Pitale, Schmidt [17]. Let S ∈ M 2 (F ) be a symmetric matrix such that d = disc S = −4 det S = 0. For and denote by F (ξ) a two-dimensional F -algebra generated by 1 2 and ξ. Note that Depending whether d is a square in F × or not, The determinant map on F (ξ) corresponds to the norm map on L, defined by N L/F (z) = zz, where z →z is the usual involution on L fixing F . We define the Legendre symbol as If L is a field, denote by ø L , p L , ̟ L the ring of integers, the maximal ideal of ø L and a fixed choice uniformizer in ø L , correspondingly. If L = F ⊕ F , let ø L = ø ⊕ ø, ̟ L = (̟, 1). Define an ideal P := pø L in ø L ; note that P = p L is prime only if It is not hard to verify that T (F ) = F (ξ) × , so that T (F ) ∼ = L × . We identify T (F ) with L × via (5). We can consider T as a subgroup of G via Let us denote by U the subgroup of G defined by and finally let R be the subgroup of G defined by R = T U .
Let Λ be a character of T (F ) such that Λ| F × = 1. Denote by Λ ⊗ θ the character of R(F ) defined by (Λ ⊗ θ)(tu) = Λ(t)θ(u) for t ∈ T (F ), u ∈ U (F ). Let π be an irreducible admissible representation of the group G(F ) with trivial central character. We say that such a π has a local Bessel model of type (Λ, θ) if π is isomorphic to a subrepresentation of the space of all locally constant functions B on G(F ) satisfying the local Bessel transformation property B(rg) = (Λ ⊗ θ)(r)B(g) for all r ∈ R(F ) and g ∈ G(F ) .
It is known by [13], [20] that if a local Bessel model exists, then it is unique. If the local Bessel model for π exists, we denote it by B π Λ,θ . In this case, we fix a (unique up to scalar) isomorphism of representations π → B π Λ,θ and denote the image of any φ ∈ π by B φ .
In the Lemma below we explain how to switch between Bessel models defined with respect to different matrices S. Together with [18, Lemma 1.1] that will allow us to assume, without any loss of generality, that the entries a, b, c and the discriminant d = b 2 − 4ac of S satisfy the following conditions: Lemma 3.1. Let S ∈ M 2 (F ) be symmetric, and let Λ be a character of the as- defines a character of T S ′ (F ). Let π be an irreducible admissible representation of G(F ). Then π has a local Bessel model of type (Λ, θ S ) if and only if it has a local Bessel model of type (Λ ′ , θ S ′ ).
Proof. Indeed, if B ∈ B π Λ,θ , then B ′ (g) := B( A α −1 t A −1 g), g ∈ G(F ) satisfies the Bessel transformation property and the map B → B ′ gives rise to a local Bessel model B π Λ ′ ,θ S ′ . 3.2. Sugano's formula. We now investigate more closely the case when π is spherical, that is, π has a non-zero G(ø)-invariant vector. Such a representation is a constituent of a representation parabolically induced from an unramified character γ of the Borel subgroup of G(F ). The values of the character γ at the matrices are called the Satake parameters of π and determine the isomorphism class of π.
Because central character of π is trivial, we can call them in turn α, β, α −1 , β −1 . Throughout this section we assume the following: (i) π is a spherical representation of G(F ), with a, b, c satisfying the conditions (7), for some non-negative integer n. The next Lemma shows equivalent ways of writing the group T (n). Thanks to this, our definition coincides with the one used in [15] and [17].
Lemma 3.2. The group T (n) defined above is isomorphic to each of the following: Therefore by the identification (5), Hence, under the assumptions (7) and via the isomorphism We already know that g must be of the form x1 2 + y ( c −a −b ) with x, y ∈ ø. However, because c ∈ ø × , we have y ∈ p n ø, and thus g = ( x x ) mod p n , which means that g ∈ T (n). The other inclusions are clear.
To prove the last assertion we use the isomorphism (⋆). Because ø × is compact and Definition 3.1. The smallest integer n for which Λ is T (n)-invariant or, equivalently, will be denoted by c(Λ).
Under the assumptions (i)-(iv), π has a local Bessel model of type (Λ, θ) for Λ as specified in Table 1 above. For example, if π is an irreducible spherical principal series representation (type I), such a local Bessel model exists for all Λ. This model contains a unique (up to multiples) G(ø)-invariant vector, which we will denote by B π . Thanks to Sugano, we have an explicit formula for the values of B Theorem 3.1 (Sugano; [25]). Assume (i)-(iv), and let Λ be such that the local Bessel model B π Λ,θ exists. Then where P (x), Q(y) and H(x, y) are (explicit) polynomials depending on α, ,L, F, Λ.
Because of the above theorem, if the local Bessel model B π Λ,θ exists, we can and will henceforth normalize B 3.3. Global Bessel models for GSp 4 . Let d be a fundamental discriminant and .
Let groups T, U, R be as above and let ψ be a fixed non-trivial character of A/Q. We define the character θ = θ S on U (A) by θ(u(X)) = ψ(tr (SX)). Let Λ be a character of T (A)/T (Q) such that Λ| A × = 1, and denote by Λ ⊗ θ the character of R(A).
Let π be an irreducible automorphic cuspidal representation of G(A) with trivial central character and V π be its space of automorphic forms. For Φ ∈ V π , we define a function B Φ on G(A) by The C-vector space of functions on G(A) spanned by {B Φ : Φ ∈ V π } is called the global Bessel space of type (Λ, θ) for π, and its vectors are called Bessel periods; it is invariant under the regular action of G(A), and when the space is non-zero, the corresponding representation is a model of π, which we call a global Bessel model of type (Λ, θ).
For Φ ∈ V π and a symmetric matrix S ∈ M 2 (Q), we define the Fourier coefficient For brevity we will often shorten Φ S,ψ to Φ S .

Ray class groups and Γ 0 (N )-equivalence
In the next section we will describe a relation between Fourier coefficients of Siegel modular forms of degree 2 and values of Bessel periods at the matrices similar to h p (l, m). As we shall see, these values are equal to sums of Fourier coefficients indexed via elements of a ray class group. The question is whether all the coefficients of fixed content and discriminant, up to equivalence, occur in such sum. It will be the subject of this section.
Recall that each coefficient a(F, T ) may be characterised according to content and discriminant of T , and each discriminant may be written as dM 2 L 2 , where d is a fundamental discriminant 3 and L content of the matrix T . From now on d will denote a negative fundamental discriminant.
In view of the relation (4) satisfied by Fourier coefficients of F , we define the set It is well-known that when M = N = 1, the set H(d, L; Γ 0 (1)) is isomorphic to the ideal class group of Q( √ d). As we shall see, when M, N > 1 the situation is more complicated. In [15], Pitale, Saha and Schmidt found a bijection between H(dM 2 , L; Γ 0 (1)) and a certain ray class group of Q( √ d), which we will call later Cl d (M ). We are going to extend their result to N > 1.
Because of the isomorphism described in Lemma 3.2, we may view Cl d (N ) as a ray class group of Q( √ d). Basing on the argument of [15], we will now describe a certain map from Cl d (N ′ ) to H(dM 2 , L; Γ 0 (N )), where N ′ is any integer divisible by M N .
Let c ∈ Cl d (N ′ ) and let t c ∈ T (A) be a representative for c such that t c ∈ p<∞ T (Q p ). By strong approximation we can write t c = γ c m c κ c , where γ c ∈ GL 2 (Q), m c ∈ GL 2 (R) + and κ c ∈ K * N ′ . Also, denote by (γ c ) f the finite part of γ c when considered as an element of GL 2 (A), thus we have the equality (γ c ) f = γ c m c , as elements of GL 2 (A). Let it is a positive definite, half-integral, symmetric matrix of discriminant d and content 1 (cf. [6], p. 209). Put Note that the matrices φ L,M (c) constructed above are not uniquely defined, as they depend on the choice of t c and κ c . However, the definition is correct for Γ 0 (N )-equivalence classes. Proof. This follows almost immediately from the proof of Proposition 5.3, [15]. The first part goes without any change. To show injectivity, it suffices to exchange a group SL 2 (Z) occurring in the second part of the proof with Γ 0 (N ). More precisely, if we assume that there exists a matrix A ∈ Γ 0 (N ) such that t Aφ L,M (c 2 )A = φ L,M (c 1 ), then A must be, in fact, an element of Γ 0 (N ) ∩ Γ 0 (M ). Observe that , and so if γ 1 , γ 2 correspond to S c1 , S c2 via (9), then γ 2 Rγ −1 1 ∈ T (Q). Therefore, if we take t 1 = γ 1 γ −1 1,∞ κ 1 and t 2 = γ 2 γ −1 2,∞ κ 2 as representatives in p<∞ T (Q p ) of c 1 and c 2 , then be the natural projection. Then the following diagram is commutative: Proof. This follows by construction. Let c ∈ Cl d (N ′ 1 ) and c 1 , c 2 , . . . , c t be the elements of Cl d (N ′ 2 ) that the map ρ sends to c. Choose distinct i, j ∈ {1, 2, . . . , t}. We will show that φ L,M (c i ) and φ L,M (c j ) are Γ 0 (N )-equivalent. For this it suffices to find γ ∈ Γ 0 (N ) such that γ ci ( M 1 ) = γ cj ( M 1 ) γ. Denote by (γ ci ) p the image of γ ci in GL 2 (Z p ), when embedded diagonally. Since c i , c j map to c, (γ ci ) p T (ord p (N ′ 1 )) = (γ cj ) p T (ord p (N ′ 1 )) for all primes p|N ′ 1 or p ∤ Hence, for each of those primes there exists g p ∈ T (ord p (N ′ 1 )) such that (γ ci ) p = (γ cj ) p g p . Note that we can choose γ ci and γ cj in such a way that Hence, for primes p| . This shows that g := γ −1 cj γ ci ∈ Γ 0 (M N ). Now it's easy to check that γ := 1/M 1 g ( M 1 ) gives a desired Γ 0 (N )-equivalence. Remark. The mapφ L,M from Cl d (M N ) to H 1 (dM 2 , L; Γ 0 (N )) is a bijection. In this way the set H 1 (dM 2 , L; Γ 0 (N )) acquires a natural group structure that makes it isomorphic to Cl d (M N ). Hence if Λ is any character of Cl d (M N ), then we can naturally think of Λ as a character of H 1 (dM 2 , L; Γ 0 (N )).
In the next section we will naturally encounter sums like for a character Λ of Cl d (N ′ ). Observe that, if we denote by ρ a natural projection from Cl d (N ′ ) to Cl d (M N ), we have the following useful fact: where C(Λ) := p p c(Λp) is the smallest integer such that Λ| T C(Λ) = 1.
Let us now try to describe more accurately the set H 1 (dM 2 , L; Γ 0 (N )). This will give us information on the coefficients occurring in the sum above. , then E(S ′ )={±1 2 , ± 1 2 1 2 + ξ S ′ , ± − 1 2 1 2 + ξ S ′ }. Proof. Note that E(S ′ ) = {g ∈ T S ′ (Q) ∩ SL 2 (Z) : det g = 1} and it doesn't depend on the content of S ′ . We may assume then that L = 1. A discussion at the beginning of section 3.1 applies also when disc S ′ = dM 2 and F = Q, i.e. there is an identification .
Therefore E(S ′ ) corresponds to the units of the ring of integers of Q( It is easy to check that they are of the form proposed above. Proposition 4.4. Suppose that S 1 , . . . , S t are matrices which comprise a complete set of (distinct) representatives for H(dM 2 , L; Γ 0 (1)), and A 1 ,..., A r form a complete set of (distinct) representatives for SL 2 (Z)/Γ 0 (N ).
(1) Assume that either d = −4, −3 or M > 1. Then A t i S j A i gives a complete set of distinct representatives for H(dM 2 , L; Γ 0 (N )), i.e.  Each equivalence class in H(dM 2 , L; Γ 0 (1)) (i.e. j ∈ {1, . . . , t} is fixed) can be written as a union of sets { t g t A i S j A i g : g ∈ Γ 0 (N )} with i ∈ {1, . . . , r}. The question is whether they are all disjoint. Assume this is not the case for the sets corresponding to i 1 and i 2 , i.e. assume there exists g ∈ Γ 0 (N ) i1 ∈ E(S j ) and Lemma 4.3 tells us precisely what these elements may be. The question is whether it does not imply i 1 = i 2 and how often this is the case.
Let us check whether the remaining cases may happen when i 1 = i 2 . Without loss of generality we may assume that L = 1. Observe that both H(−4, 1; Γ 0 (1)) and H(−3, 1; Γ 0 (1)) contain only one class, namely the one determined by 1 2 and can be written uniquely in a reduced form, that is with |b ′ | ≤ a ′ ≤ c ′ (e.g. Choose a set of representatives for SL 2 (Z)/Γ 0 (N ) to be 4 Hence, if we fix u, v, then u ′ , v ′ are uniquely determined and thus there are We will look for the solutions to the first condition, the latter one being symmetric. From similar reasons as in the previous situation, u ′ = vu ′ and gcd(u ′ , N ) = 1. Hence, gcd(v, v ′ )|uu ′ , and thus v and v ′ are coprime. Our condition becomes N v |v ′ + u ′ (u + v) and implies v ′ = gcd(N, u + v). Let t = u + v and write t = v ′ t. It is easy to see that t runs through the rests modulo N/v and gcd(t, v) = 1. Moreover, N vv ′ |1 + u ′ t. Hence, if we fix v and u, v ′ and u ′ are uniquely determined. Similarly, if N |uũ + vṽ + uṽ, thenũ andṽ are uniquely determined by u, v. Moreover, it is easy to check that the conditions (⋆⋆) hold at the same time only if v = v ′ = 1 and u = u ′ satisfies u 2 + u + 1 ≡ 0 (mod N ). Hence the conditions (⋆⋆) and uniqueness of the solution for each of them imply that unless v = 1 and u 2 + u + 1 ≡ 0 (mod N ), the matrix ( * u * v ) is Γ 0 (N )-equivalent to exactly two matrices. Therefore, there are N 3 Γ 0 (N )-non-equivalent classes within each class in H(−3, L; Γ 0 (1)), where In the following lemma we compute the quantities L −4 and L −3 , and that finishes the proof. Proof. This follows from Chinese Remainder Theorem and two basic facts: • For an odd prime p, (Z/p n Z) × is a cyclic group of order p n−1 (p − 1).
• (Z/2 n Z) × is cyclic of order 1 and 2 for n = 1 and 2, respectively. If n ≥ 3, then it is the product of two cyclic groups, one of order 2, the other of order 2 n−2 .
If p ≡ −1 (mod 6), then the order of (Z/p n Z) × is not divisible by 3. In the other case, there are two elements of order 3, u 0 and u 2 0 , say. Both of them are zeros of the polynomial u 2 + u + 1 = 0 mod p n .

The relation
Fix Φ ∈ V π , N ∈ N and assume that Φ is right invariant by the subgroup I N of G(A f ). • the conductor of ψ p is Z p for all (finite) primes p, so that θ(u(X)) := ψ(tr (SX)) is a character of U (A). Let Λ = p≤∞ Λ p be a character of T (A)/T (Q)T (R) such that Λ| A × = 1, and let C(Λ) = p p c(Λp) be the smallest integer such that Λ| T C(Λ) = 1 (cf. Def. 3.1, 4.1). Suppose that Φ =Φ ⊗ p∤N2 φ p is a pure tensor in the space of π away from the level. If π has a global Bessel model of type (Λ, θ), then for each place p of Q, π p has a local Bessel model of type (Λ p , θ p ) and each φ p corresponds to a (unique up to multiples) vector B φp in the local Bessel model of π p . Remark. If π p is a spherical representation with trivial central character and it is not of type I (that is, it is of type IIb, IIIb, IVd, Vd or VId), then π p admits a local Bessel model if and only if Λ = 1, in which case c(Λ p ) = 0. The only representations that will occur in the proof of main theorem regarding Maass relations are of type IIb, and thus in what follows we could take simply M Φ,Λ = 1.
The following lemma is the base for our main results. . If a local (Λ p , θ p )-Bessel model exists at all primes p ∤ N 2 , then the following relation holds: l, m)) . B φp (h p (l, m)) .
Using this equation with L ′ , M ′ as in the statement of the lemma, we obtain the relation (11). Note that without the assumption M Φ,Λ |M ′ the statement is still true, but we have zeros on both sides of the equality.
From the simple looking relation (11) we obtain a correspondence between the Fourier coefficients (8) that will be crucial for our applications. We start with an auxiliary lemma.
and let γ f be the image of γ in G(A f ). Then for any automorphic form Φ on G(A), any matrix T ∈ M sym 2 (Q) and g ∞ ∈ G(R) + we have Using the fact that Φ is left G(Q)-invariant and the substitution X = α −1 AY t A, we obtain Lemma 5.3. Let Φ be an automorphic form on G(A) that satisfies the equation (2), and let F be as in (3). Then for any matrix T ∈ M sym 2 (Q) and g ∞ ∈ G(R) Proof. It is easy to check that the automorphy factor j has the property j(g 1 g 2 , Z) = j(g 1 , g 2 · Z)j(g 2 , Z) . Hence, e(−tr (T X))e(tr (T ′ (g ∞ · i 2 )))e(tr (T ′ X))dX = µ(g ∞ ) k j(g ∞ , i 2 ) −k a(F, T )e(tr (T (g ∞ · i 2 ))) .
Proof. In view of the relation (11) ∈ G(R) + . In particular, Φ L,M is right invariant by T MN1 and K * MN1 . Following the notation of section 4, we can write where we choose t c ∈ p<∞ T (Q p ), and by strong approximation theorem write t c = γ c m c κ c with γ c ∈ GL 2 (Q), m c ∈ GL 2 (R) + and κ c ∈ K * MN1 . With this preparation we are ready to compute the values B Φ (H(L, M )).
Observe that if C(Λ) ∤ M N 1 , then the inner integral is equal to zero and the equation (12) holds. Henceforth we assume that C(Λ)|M N 1 . With this assumption and using Lemma 5.2 twice, we have where φ L,M (c) is defined as in (10). Further, . Then for all l, m)) .
Proof. This follows immediately from Theorem 5.1 and Proposition 3.11, [22], which states that in our setting the following conditions are equivalent: 5 It is enough to assume that F is an eigenform of the Hecke operators T (p) and T (p 2 ) at p ∤ N 2 . For the definition of these Hecke operators see for example [8] or [1].
(i) F is an eigenform of the local Hecke algebra at all primes p ∤ N 2 .
(ii) If π ′ , π ′′ are two irreducible cuspidal representations both of which occur as subrepresentations of the representation π associated with F , then π ′ p ∼ = π ′′ p for all primes p ∤ N 2 . As a result, F = i F i , where each F i has the same local data at p ∤ N 2 and is as in Theorem 5.1.

Maass relations
Let F be a cuspidal Siegel modular form invariant under the action of Γ 0 (N 1 , N 2 ) that is an eigenform of the local Hecke algebra at primes p ∤ N 2 . Let Φ be the adelisation of F , and π = ⊗ p π p the associated automorphic representation. Suppose that for primes p ∤ N 2 , π p = χ p 1 GL(2) ⋊ χ −1 p with an unramified character χ p of Q × p (a representation of type IIb according to Table 1). Note that these are nontempered, non-generic representations. The set of π obtained in this way is precisely the set of CAP representations attached to the Siegel parabolic subgroup of G(A) (cf. [4]).
Lemma 6.1. For representation π as above, any vectorΦ = ⊗ pφp in the vector space V π of π and any non-degenerate matrix S ∈ M sym 2 (Q), we have: Proof. LetΦ = ⊗ pφp be as in the lemma, and let S = {p : p|N 2 }. Without loss of generality we may assume g = 1 4 . Let V S be the subspace of V π generated by all vectors of the form ⊗ p∈Sφp ⊗ p / ∈S ψ p with ψ p ∈ V πp . The right action of ⊗ p / ∈S G(Q p ) on V S makes V S a representation isomorphic to ⊗ p / ∈S π p (we put here Q ∞ := R). Define Note that β(π(t)Ψ) = Ψ S (t). We need to show that β(π(t)Φ) = β(Φ) for all t ∈ p / ∈S T (Q p ). This is trivial if β ≡ 0. So assume β ≡ 0. Let For each place p / ∈ S we get a functional β p on V πp via . Then β p (φ ′ p ) = 0 and thus β p is a non-zero functional on V πp . Clearly, β p satisfies β p (π p (u)ψ p ) = θ S (u)β p (ψ p ) for all ψ p ∈ V πp and u ∈ U (Q p ) By [16,Corollary 4.2], the matrix S satisfies the conditions of [15,Lemma 4.1], and therefore by this lemma • the space of such functionals β p is one-dimensional, • β p (π p (t)ψ p ) = β p (ψ p ) for all ψ p ∈ V πp and t ∈ T (Q p ). So there exists a constant C S such that whenever Ψ ∈ V S corresponds to ⊗ p∈Sφp ⊗ p / ∈S ψ p . Hence β(π(t)Φ) = β(Φ) for all t ∈ p / ∈S T (Q p ). Lemma 6.2. Let F, N 1 be as above, S = S(d), and L, M any positive integers. Then for any c 1 , c 2 ∈ Cl d (M N 1 ), a(F, L ( M 1 ) S c1 ( M 1 )) = a(F, L ( M 1 ) S c2 ( M 1 )) . Proof. Let {t c } c be a set of representatives of Cl d (M N 1 ). We may choose t c so that t c,p = 1 2 for all p|N 2 . Indeed, ift ∈ T (Q) is such thatt p = t c,p for all p|N 2 , then t c =t p|N2 1 2 p∤N2t −1 p t c,p . From the proof of Theorem 5.1, and using its notation, we get does not depend on c.
Hence, it makes sense to write a(F ; dM 2 , L) for any Fourier coefficient of F that is of the form a(F, L ( M 1 ) S c ( M 1 )) for some c ∈ Cl d (M N 1 ), or in another words, for a(F, T ) with T ∈ H 1 (dM 2 , L; Γ 0 (N 1 )).
The following theorem generalises [15,Theorem 5.1] to cuspidal Siegel modular forms of level Γ 0 (N 1 , N 2 ) with N 2 > 1. B φp (h p (l p , m p )) , 6 Since at p ∤ N 2 , πp = χp1 GL(2) ⋊ χ −1 p is a representation of type IIb, so according to Table 1  Note that the last product can actually be taken over all primes p|LM that do not divide N 2 . Indeed, in the product over p|LM/r, p ∤ N 2 we miss only those places p for which both r p = l p and m p = 0. But in this case B φp (h p (0, l p + m p − r p )) = B φp (h p (0, 0)) = 1 by Theorem 3.1. Moreover, it is known [15, Theorem 2.1] that the spherical vectors of the representations of type IIb satisfy the equation B φp (h p (0, l p + m p − r p )) , which is true.
It is unfortunate that our method gives us an access only to the coefficients of the form (15), which are not yet fully characterised. Nevertheless, their partial study in section 4 with subsequent Corollary 4.1 already leads to a satisfactory result which generalises the full level case: Remark. One of the main differences between the classical Maass relations (1) and the Maass relations (14) is that the coefficients a(F, T ) on the right hand side of the first equality have the matrix T of content 1, and in particular, the (2, 2)-entry of T equals 1. As we pointed out in the introduction, Saito-Kurokawa lifts do not always enjoy this property. This is the case for example when a lift is a paramodular form. Even though our result does not apply to paramodular forms (as representations of type IIb are not generic), we treat this fact as an indication that a further study of Fourier coefficients of Siegel modular forms of higher levels might be necessary to distinguish the cases where our result could be improved.